There are more than 50 graveyards in Scottsdale, Arizona, a desert town whose original Indian name translates as "rotting hay". Some are standard cemeteries, some are said to be a stomping ground for ghosts and ghouls – and, around the turn of the century, a number of them played host to Emma Stone.

She is 25 now; back then, she says, she was slighter, more waifish and "somewhat creepy. I was always in cemeteries when I was a kid," she says, her voice gravelly. "I was really into spooky things like ghosts and death. I have always had an awareness of mortality. It makes you live more fully if you are aware that you only have a finite amount of time."

Such thinking has served her well. She left high school early, before moving to Los Angeles at the age of 15, after putting together a PowerPoint presentation to persuade her parents that relocating to tinseltown would be the best thing for everyone. You only live once, after all.

Stone (left) in The Help, 2011. Photograph: Dale Robinette/Unit

"I have had an awareness of mortality since I was around six or seven," she says. "If it's not been in the forefront of my mind, it has at least been creeping over my shoulders." We are talking in a hotel room in LA, though she now lives in New York, where she enjoys popping in to Brooklyn's Morbid Anatomy Library, especially for the "little foetal pigs in jars".

She smiles. "There is something oddly comforting about death. I have never been an entirely carefree person." Her parents are of a similar mindset; they sanctioned the move to LA "so I could pursue what I wanted to do as quickly as possible. Everything happened younger because of that mentality; it has made me appreciate life in a really big way. Any time I lose sight of that and I feel like I am bulletproof, that's when I get into trouble."

Stone as high school student Olive Penderghast in Easy A, 2010. Photograph: Adam Taylor/CTMG

Her sense of invincibility is understandable. She is promoting The Amazing Spider-Man 2; its predecessor took £450m worldwide. Her rise has been speedy and bump-free – no wonder she seems as quick and winning in the flesh as she does on screen. "Confident? I am a mess in every way. I pulled it together today for the most part," she says, grinning.

She is sweet, but there is a sense of salt beneath it, a wisdom beyond her years. She is witty, too, a natural comedian, with an insouciance that makes her delivery feel natural, even in more serious-minded fare – as an aspiring journalist kicking against 60s segregation in The Help – or in tonally tricky roles, such as the repressed lawyer who falls for Ryan Gosling in Crazy, Stupid, Love or the plucky moll – who again falls for Gosling – in Gangster Squad.

In Spider-Man, she stars opposite real-life boyfriend Andrew Garfield, who plays Spider-Man, AKA Peter Parker; they met on the first film, flirted and played pranks, then got together. "The chemistry between those two is like old Hollywood, Hepburn and Tracy," says Marvel's studio head, Avi Arad, who, in part, credits the film's enormous success to the happy casting.

Stone agrees. "I could relate to that feeling, that all-encompassing, you'll-die-for-this-person first love. It's everything I felt in my first relationship, that total Romeo and Juliet story played out. I never had an opportunity to fall in love like that in a movie before."

You feel Stone has brought a lot of herself to the role of Gwen Stacy; her and Gwen's concerns dovetail in other ways, too. The film opens with a ticking clock and our heroine telling her fellow graduating classmates (she is off to Oxford to study science) that "Time is luck". "After what's happened to her dad [a police chief, who died in the first film], Gwen has a great awareness of mortality and time and how quickly life goes," says Stone. "She now has an emotionally driven reason to make logical decisions, and her story resonated hugely with me, because it is so incredibly dramatic.

"I was that ambitious when I was very young as well," Stone says. "Gwen is clear on what she wants to do with her life – in terms of her job and her sense of purpose. I think and hope that I am in the same ballpark there." She grins again. "Science-wise, though, she is much better than me."

Stone in the 2009 horror comedy Zombieland. Photograph: Glen Wilson

The young Emily Stone – she took the name Emma because of an Equity clash – grew up on the Camelback Club golf course in Arizona, which her parents owned.

"My parents always understood who I was," she says, "and they made sure that I never wanted to lie to them." The mantra they hammered home was that she would never get into trouble for telling the truth. "I could do anything as long as I did not lie. They wouldn't punish me if I was honest, and that really worked in our home."

She caught the acting bug from early exposure to Steve Martin films such as The Jerk and Planes, Trains and Automobiles and, significantly, started off in a children's improvisation class. Early heroes included Diane Keaton, Gilda Radner and Lucille Ball. "But mostly I loved the Spice Girls," she says, "especially Baby. I think it was all that girl power and the platform shoes and pigtails."

It was not too long after that she switched on the projector, delivered that presentation to her parents, and secured the move to Los Angeles with her mother, starting out in television and appearing in the likes of Medium and Malcolm in the Middle.

Her feature film debut was auspicious and striking – she played the sassy buddy of Jonah Hill in Superbad – and rapidly followed it with roles in The Rocker and The House Bunny. There was a blip, in 2009, with the critically reviled Matthew McConaughey rom-com Ghosts of Girlfriends Past, but then back on track for horror-comedy Zombieland and quirky indie film Paper Man.

Stone with Ryan Gosling in Crazy, Stupid, Love, 2011. Photograph: Warner Brothers/Sportsphoto Ltd/Allstar

But it was 2010's Easy A that proved her launchpad. The film – a slightly filthy high school comedy influenced by The Scarlet Letter – didn't get a huge release in the UK but was a sleeper hit in the US (taking $58m from an $8m budget), and produced a ticker-tape of good notices for Stone. Richard Corliss of Time magazine called her performance one of the top 10 of the year; Roger Ebert said it made her a star; John Griffiths from Us Weekly praised her "husky voice and fiery hair" and likened her to Lindsay Lohan.

"It was pivotal," she says. "I had no idea it would do so much for me." Then came Crazy, Stupid, Love and The Help, Spider-Man and its sequel. Next up is Alejandro González Iñárritu's Birdman and Magic in the Moonlight, the new Woody Allen film, set in the 1920s, in which she is romanced by Colin Firth. "Working with Woody was very different from what I expected," she says. "I was expecting him to be very removed and very quiet as a director. But he was incredibly funny and expressive."

She is currently shooting a Cameron Crowe comedy in which she plays a fighter pilot. After that comes a reunion with Easy A's director Will Gluck – as yet untitled, but she is listed as an executive producer as well as star. "You want to remain authentic but also want to retain some semblance of a private life. That's what I struggle with the most," she says.

Stone as Grace Faraday in the 2013 noir-ish LA drama Gangster Squad. Photograph: Wilson Webb/AP

Her life has changed considerably since the first Spider-Man film. "More work has come my way and it has compromised my anonymity. That loss is weird; I get to hear my name a lot more: 'Emma! Emma!' It's a bit odd. Different days I react differently, but it is still so foreign to me that someone could walk by and put a phone in my face as if I am not another person. I am not going to punch anybody. I'm not going to scream and rip someone's face off.

"Fortunately, it's not that people are camped outside my house, living there, waiting to get a picture. My life is not unliveable. I get to stay in wonderful places and travel first class. It feels very extravagant. It's very cool to have that experience. But you also realise that life is not about that at all once you are there. It's like, 'Wow, a big, empty house!'"

If she gets lonely, of course, she could always take a walk around the local graveyard. It's not hard to picture her, dodging the autograph-hunters, wisecracking at the tombstones, seizing life while she can.

• The Amazing Spider-Man 2 is released in the UK on 18 April.