Chris Martin is sitting in a Las Vegas hotel, thinking about the power of songs. Specifically, he is thinking about something Bono once said about Coldplay - that they have "a sense that the world's hardest of hearts can be thawed by a great melody". Martin disagrees. "I think there are some hearts that are fucked," he says.

Then, as he often does, he changes his mind. "Actually, to be fair, I was wondering whether certain people's policies would change if they heard certain songs," he says, in his slightly sinusy, barely West Country-accented voice. "Would it really be possible to start Nazi Germany if you'd just been listening to Bob Marley's Exodus back-to-back for the past three weeks and getting stoned? Would the idea of the Holocaust seem so appealing? I know this sounds really trite, but I mean it seriously, because music is something that every human responds to. There's a reason why people who've had bad relationships with their parents listen to angry stuff."

Martin, who has a very good relationship with his parents, tends to listen to - and write - dreamy, idealistic stuff. Coldplay's 28-year-old frontman is culturally engaged and politically passionate. The band give 10% of their considerable earnings to charity, although they don't like to talk about it, even to say which organisations benefit. Martin has travelled with Oxfam to Ghana and Haiti, and personally lobbied the World Trade Organisation at its 2003 meeting in Mexico. He doesn't drink alcohol or coffee, but when Coldplay were making their first album, Parachutes, he forced himself to booze to the point of vomiting as penance for briefly kicking drummer Will Champion out of the band. He wrote one of Coldplay's hits, Trouble, while huddled in a recording studio cupboard. He is a sensitive soul.

"So in some sense," he continues, "I do think melodies can do a lot. It would be interesting to see how the world would be different if Dick Cheney really listened to Radiohead's OK Computer. I think the world would probably improve. That album is fucking brilliant. It changed my life, so why wouldn't it change his?"

Martin needs Coldplay's long-awaited new album, X&Y, to have a similarly profound effect. The band's last album, 2002's A Rush Of Blood To The Head, did well enough, selling 10.5m copies, but their third has to accomplish more - it has to be the band's "definitive" album. It has taken Martin, Champion, guitarist Jonny Buckland and bass player Guy Berryman 18 months - in eight studios, in five cities, in two countries - to make it. They started, then stopped; they chucked stuff out, and started again. Along the way, they took advice from Michael Stipe, resurrected a song originally written for Johnny Cash, sought inspiration from a keyboard that once belonged to Gwyneth Paltrow's father, wrote a fan letter to Kraftwerk, sampled a 17th-century hymn and, in a sign of what a big deal a Coldplay album now is, saw their tardiness blamed for a fall in their record company's share price. (This annoyed Martin, who told reporters, "I don't really care about EMI. I think shareholders are the great evil of this modern world.")

Now the hard work is done, the really hard work begins: taking the record around the world, starting here in Las Vegas. Martin looks around him and says he always feels phoney in Vegas: the cheesy, tropical vibes of the Hard Rock hotel make him think of TV's Fantasy Island, or of cabaret singer Tony Ferrino, one of Steve Coogan's less successful creations. "I think that was Coogan's X&Y," Martin says with a grin. "He was very proud of it, but no one else was."

Behind the hotel is a man-made beach - a perfect confection of water, sand, rocks, palms, bikinis and bermuda shorts. Silly muscles and fake breasts jostle for space in the late afternoon desert heat. Cocktails are being served to sunloungers and rock music pounds from the speakers. Martin pushes back the double glass doors and walks out into the blinding sunshine to take it all in. He doesn't get it, he says. Here he is, surrounded by the beautiful people, supposedly one of the biggest rock stars in the world - so why doesn't he feel as cool as they do? He is dressed all in black, with a scarf wound around his neck. There are smears of white cream crusting on his ears, and an unfortunate boil on his left temple. "I do an hour's yoga and go running every day. Then I see a picture of myself and I still look like a skinny, potbellied idiot - and I thought I had turned into this superhunk!"

Earlier that week, Speed Of Sound, the first single from X&Y, entered the Billboard chart at No 8, making Coldplay the first British band to debut in the US top 10 since the Beatles in 1968. This weekend, they are headlining at the massive Coachella music festival, the closest thing America has to Glastonbury. If Coldplay are occasionally derided in the UK as glum, overly sincere rockers, surely Martin recognises that in the US right now they are dead cool? "I don't think we're considered cool anywhere."

Tonight, Coldplay are performing in the hotel's 1,500-capacity venue, the Joint. The gig will be a warm-up for the first real unveiling of X&Y: in 30 hours the band will be out in the California desert, playing to a crowd of 50,000. In the meantime, Martin wants to buy me some sunglasses, to make up for getting my name wrong earlier. ("Have I offended you? Does the backlash start now?") On our way to a beachside kiosk, we are intercepted by an English stag party, pickled in the sunshine. It's the singer from Coldplay! Any chance of a photo? "C'mon then," says Martin, flinging his arms around their shoulders and crouching his 6ft 2in frame down to their height. Kelly, the singer's as-wide-as-he-is-tall security detail, stands impassively by.

Then there is another distraction, the actor Daniel Craig gingerly trying to say hello. He filmed Sylvia with Gwyneth Paltrow, and has just finished making another movie with her, the Truman Capote drama Every Word Is True. Now he is in town to promote the US release of the British gangster flick Layer Cake. They stop and talk, and afterwards Martin tells me Craig is the new James Bond. Really, I say? Officially, no decision has been made. "Oh," says Martin, and mutters something about not knowing after all.

This is the tension between the two sides of Martin's life, the X and the Y. He is a people's songwriter, an unashamedly mainstream people's songwriter, a normal, bumbling young man who, by dint of being married to Paltrow, has somehow become a member of fame's superleague. He is very aware of this. He still can't believe people actually like Coldplay, let alone that Steven Spielberg is now his godfather-in-law. "In my darker days," he tells me later, "I think people just see us as the band who married into Hollywood."

Right now, though, he is jabbering away to the girl at the kiosk, bouncing on his heels. We pick out some sunglasses, which Martin insists on paying for, and then head for a cabana at the back of the beach. He orders a strawberry daiquiri for me and a virgin daiquiri for his pure, toxin-free self. He is very chipper. "It's incredible," he says, wide-eyed. "We've started now. Our whole life at the moment is like Magnus Magnusson at the end with the clock - we've just got to get on with it." You've started so you'll finish? "Exactly. It's just been all cylinders. I mean, I said to" - he pauses as he remembers not to mention someone (Paltrow?) by name - "my friend yesterday, I still can't believe that I'm allowed to do it. That I got given this thing. I really can't believe it."

Martin's insecurity is a funny thing. He simply can't fathom that a band with him in it currently outsells Radiohead, U2, REM and Oasis. "I'm thinking, God, I remember being 13 and reading about U2's Achtung Baby, or 18 and reading about Oasis's Definitely Maybe. And now we're hang -" He stops himself again: "hanging out" sounds too naff. So he simply says he finds it "hilarious" that Coldplay are talked of in the same sentence as bands whose records he rushed out to buy when he was 15. "It just reminds me that my dreams have come true on a very pure level."

He is aware that with fame comes influence, and he has used it to champion the causes he holds dear (Fair Trade, Oxfam, the Make Poverty History campaign), even though he knows that the figure of the "conscious rocker" is open to ridicule. Liam Gallagher, noting Martin's propensity for writing slogans on the back of his hand, once called him a "knobhead student". (Martin half agrees: "In a sense, he's right. I am a knobhead. But I'm not a student any more.") Yet if you were to put it to Martin that he might be the next Bono, he would run a mile. Put a microphone in front of U2's frontman and he will orate like a statesman. Faced with a crowd, or even a lone interviewer, Martin waffles like your clever, amusing but slightly confused mate. He does not think he's the next messiah.

At the soundcheck that afternoon, Martin playfully rambles through bits of Oasis's Champagne Supernova, REM's Perfect Circle and the Troggs' Wild Thing. The white stuff on his ears, it turns out, is monitor cream, used to fix his tiny onstage headphones. He is looking forward to headlining the middle night at Glastonbury, he says; if he is at all fazed by the prospect of this summer's huge UK stadium shows, he isn't showing it.

Where Coldplay used to wear what might charitably be described as student chic, they now dress like rock stars: black tops, black trousers, white shoes."There's great security in looking over at Jonny and seeing he's wearing the same coloured shoes as me," he says. "I suppose it's the same reason the army wears a uniform - so that you feel part of a clan. And when we're all dressed that way, I just feel very much like, it's OK, coz I'm part of this team."

From the start, Coldplay have been very much a foursome, a tight partnership of friends and equals. (Martin is the main songwriter, but each receives a 25% share of earnings.) If their bonds were weaker, they might not have survived the rapid ascent from indie zeroes to stadium-rock heroes. Martin, Berryman (27, from Fife via Canterbury), Buckland (27, from London via north Wales) and Champion (27, from Southampton) met in their first year at University College London, in 1996. The current line-up was formalised in late 1997, and in February 1998 they played their first gig in a Camden pub. A year later came their first, self-released EP, swiftly followed by a record deal. Pausing only to take their degrees - Martin, ancient history; Champion, anthropology; Buckland, maths and astronomy; Berryman had dropped out of engineering - they began recording their debut album. In summer 2000, Parachutes entered the chart at No 1.

Five years on, there is a lot riding on the new album, but the stress can wait. Right now, Martin is excited about everything. Earlier that week, he was in Los Angeles, where Paltrow's mother, the actor Blythe Danner, has a house. They'd had Brian Eno over for tea - Coldplay used bits of Eno's Apollo album as their intro music on their last tour. "Actually, we use everything of his everywhere, some less subtly than others. I love that man with a deep passion. He's the cleverest man in the world, since Bob Marley died and John Lennon died. He is fucking mindblowing." Martin also went to a lecture given by Eno and the inventor Danny Hillis, called Thinking After Einstein, at LA's Skirball Cultural Centre. Eno and Hillis are members of the Long Now Foundation, which "hopes to provide counterpoints to today's 'faster/cheaper' mindset and promote 'slower/better' thinking". The lecture, he says, "was just about them and their way of working. It was fascinating, actually."

Yesterday he had an enforced day off because he was ill with a bug, courtesy of one-year-old Apple Blythe Alison Martin. "I caught it off my daughter, so I don't care," he says.

Has being a dad changed him? "Well, in the same way I can't believe that I've been allowed to be in Coldplay, I can't believe I'm allowed to be Apple's dad." He takes a ruminative suck on his cocktail. "It's extraordinary. Every day I treat it like, 'Wow, so you want to hang out with me!' It's made me see things in more extreme colour. If we're going to do something with the band, I want to do it properly and to the power of 10. And if I get upset about something now, I get upset about it a lot more. So it's made things more vivid. Things that seemed threatening seem even more threatening - on a very simple level, things like environmental stuff. Who's in power. Or Fair Trade." Fatherhood, he says, "makes me more driven".

He says that while making the album he kept meeting people who had been abandoned by their fathers, "which is weird. I never really thought about that till I had Apple. The only thing I can think of is, you're so terrified about messing them up that you think you'll do less damage if you're not there."

We finish our cocktails and head back to the hotel. Martin asks if I like my new sunglasses, and says they cost $16. In fact, I later find out, they cost closer to $100. Martin is embarrassed by his efforts to make up for his earlier embarrassment.

X&Y is indeed the work of an increasingly driven, punchier band. It uses more electronic-based noises, some found by Martin on the "massive" keyboard Bruce Paltrow bought not long before he died in autumn 2002. In places, the album suggests sparkling modern reboots of indie-rock classics: on the soaring opener, Square One, it's U2 circa The Unforgettable Fire; The Hardest Part is a steal from REM's Losing My Religion, so blatant that the band almost didn't include it. More intriguing are the songs where they sound more like themselves, even if these include a bit of borrowing. Talk makes uplifting use of the riff from Kraftwerk's Computer Love, while 'Til Kingdom Come is a hymnal acoustic song the band originally recorded as a backing track for Johnny Cash, a song the Man in Black never got around to singing. Twisted Logic is an intense, angry track encouraging people to make the right decisions about how they live their lives and how they treat the planet. Martin describes it as like "being slapped by a model" - impactful, but beguilingly so. It's his most political song to date.

The album is a confident step up for the band, and will no doubt sell in the many millions. Compared with "edgier", newer British bands - Franz Ferdinand, Kasabian, Razorlight, Bloc Party - Coldplay have a more traditional sound and so a broader appeal. Critically, they often get lumped in with Athlete, Travis, Embrace and Keane, bands deemed to make polite, non-threatening, melodic rock, but Coldplay's tunes are more powerful, and Martin a more engaging frontman. This summer they will be pitched head-to-head with Oasis, whose new album, Don't Believe The Truth, is released next week. The Oasis comeback album is a convincing return to the fray, but it's still an Oasis album, chock-full of steals from the rock classicist's manual - good fun, but hardly fodder for the soul. X&Y, on the other hand, is heartfelt stuff, with thumping guitar lines, emotive piano and sublime electronic embellishments. Martin's voice, too, is a superior and astonishingly limber tool, maintained by a regime involving cinnamon, oils and hot lemon drinks.

The band's new muscularity is a long way from Parachutes, which, according to Berryman, was "a quiet, polite record". Sure, it had the anthemic Yellow, the song of the summer of 2000 and the tune that broke Coldplay. But, as Buckland admits, it could also be seen as a bit of a downer. "We were in France before the album came out. We had the album in a certain order. And every French interviewer would say to us [adopts 'Allo 'Allo accent], 'So, why are you so miserable?' We didn't think of it as a miserable album." He laughs. "But then you listened to it and it just had a really, really depressing ending."

A Rush Of Blood To The Head was more dynamic, edgier. Coldplay debuted much of it during their headline set at Glastonbury in 2002, more than two months before the album came out. No one had heard the tracks before, but the immediate power of songs such as Clocks and The Scientist, epic and intimate at the same time, was clear.

That night in Las Vegas, in the Hard Rock's small, tasteful venue - all blond wood, geometrically arrayed guitars on the wall, the po-faced legend "Humanity is instrumental" above the stage - the new songs burst into life. Down the front, Daniel Craig boogies self-consciously with a glamorous young woman. Upstairs in the balcony there is enthusiastic jewellery-rattling from Robbie Williams, Jessica Simpson, Elisabeth Shue, Matthew Perry, Courteney Cox and David Arquette. Dave Navarro and Carmen Electra are sufficiently moved to snog the faces off each other.

Backstage before showtime, Martin had been in top form, running around in snug black pants, throwing rubbish karate moves. Post-gig, it's a different story. The band had a terrible time onstage. They couldn't hear a thing. They felt the new songs stuttered in places. It wasn't their kind of crowd. So it is a rather glum Martin who makes small talk in a corridor outside the dressing rooms with his American booking agent - whose toddler daughter is also called Apple. Martin tells him that, since marrying Paltrow, whose father was Jewish, he is now an "honorary Jew" - "I just had my first Passover." Afterwards, everyone heads off to the casino with a bag of bespoke Coldplay gambling chips. Martin and Champion (whose wife is travelling with him) make their excuses early, but Berryman and Buckland play blackjack until the early hours.

The following lunchtime Coldplay fly by private jet to Palm Springs, 35 minutes from Las Vegas. The band can now afford to fly wherever possible, and the increased privacy and speed mean that Apple will be able to join her father on tour more often. "I certainly don't want her to stay at home all the time," Martin says. "As she gets older, hopefully she'll come out as and when she wants. I always thought it'd be cool to be in school and say, 'I'm not coming in today - I'm off to Costa Rica to see my dad play.' I do think that wins you a few points."

From the airport, it is a short drive to the La Quinta resort, where the paparazzi are waiting for a chance to snap Paltrow, who is joining the band from LA. We head through lush, irrigated polo fields to the edge of the desert, where the band have their pictures taken. "Our Joshua Tree fantasies come true!" jokes Martin as Berryman, Buckland and Champion stay silent behind their shades. Martin keeps thrusting his left hand towards the camera. He is wearing red and blue gaffer tape on his middle and index fingers, designed to draw attention to the Make Trade Fair "equals" sign he has drawn on the back of his hand. He never misses a chance to deliver the message.

He is not always so happy to be photographed. The paparazzi stake out his north London home almost daily, desperate for a shot of him, Paltrow or Apple, preferably all three. "People forget that my wife is just a young mother," he says. "I hate Heat magazine with a passion," he declares, before admitting, "I will pick it up at the dentist's ahead of some other magazine, coz I'm like, 'Oh, I wonder what ...' "

You wonder if you're in it? "Well, not really. But, you know, we've all got that in us." He pauses. "I don't hate Heat especially, but I hate that side of me that likes to ogle because I know what it's like to be ogled."

And even the mild-mannered can snap. Last year Martin was cautioned for common assault after tussling with a photographer who tried to take pictures of him and Paltrow leaving a London restaurant (Paltrow was seven months pregnant at the time). In 2003, he took a rock to the windscreen of a photographer who had been following the couple around a beach in Australia. He was fined for this, and says now that he regrets "any form of aggression towards anybody else, but sometimes you can't really control the animal instinct inside you. When you or someone close to you feels directly threatened or shocked, you respond. If you go up and kick a dog, it's going to turn round and bite you."

He pointedly never mentions his wife by name. He is clearly uncomfortable talking about her, but makes things difficult for himself by being unfailingly polite and incapable of telling a lie. So, while refusing to answer questions about the specifics of their relationship - meeting her at a gig in 2002; their marriage in 2003; whether Paltrow, an accomplished singer whose soundtrack to the film Duets earned her a No 1 in Australia, was his muse for the new album - he will say this about being married to a famous person: "While I really understand all the interest in it, after half an hour with anybody they just become another person to you. And you just become another person to them. Of course, both of us knew who the other was before we met. But we're just little people, little creatures, as well. It's not like if a bomb went off we wouldn't be killed."

Late on Saturday night, Coldplay finally take to the main stage at Coachella. Martin bounds on as the opening keyboard washes of Square One flood the 80 acres of the Empire Polo Grounds. Justin Timberlake and Cameron Diaz stand clapping on one side of the stage, Paltrow on the other. Familiar anthems are greeted with roars from the crowd. Martin encourages everyone to clap along during God Put A Smile Upon Your Face, saying, "I know we're foreign, but give us a chance." He sings some Weezer (who were on just before) and some Nine Inch Nails (who are headlining on Sunday). It's easy to imagine the songs from X&Y blasting, rippling and drifting out in stadiums around the world for the next 18 months. Speed Of Sound is massive.

After waffling something about everyone joining forces to try to get "the heavens to open" for Low, Martin apologises: "I don't speak particularly well. That's one of the consequences of being extremely ugly." As his smiling face looms from the giant screens (yesterday's boil has subsided), 50,000 screams begged to differ.

Around midnight, they finish with Fix You, an organ-led song about reconciliation and hope that climaxes with a rousing, gospelly chorus. The crowd love it. Martin tells them that last night Coldplay played one of their worst shows ever, "in a place that had more silicone than Silicon Valley. Tonight," he beams, "is the polar opposite."

Afterwards, in the crowded paddock outside the band's backstage caravan, there is a sense of a job well done. Berryman, Buckland and Champion crack open cans of lager. Paltrow raids a bucket of cold drinks. The huge lumps of ginger provided are as yet untouched, but they'll soon be needing more alcohol. Management, crew, people from the British and American record labels and the baby-faced boy-man from Keane mill around. Martin pinballs from guest to guest, an exuberant host. Coldplay have their first big gig under their belt. Their frontman can't help worrying what the world will make of X&Y, but officially the suspense is over; they are on the road. Tomorrow they play football at Robbie Williams's house in LA. Life is good.

Eventually Martin makes his excuses. "Sorry," he says, "got to go and find my missus."

· X&Y is released on June 6