There were many "really stupid, brilliant films" played on the VHS player in Andrew Garfield's house when he was young. Teen Wolf, the Wayne's World movies, Back to the Future and Bill and Ted's pair of adventures were favourites. So were boxing stories. A clutch of heroes could be seen swinging their fists on screen — Stallone, De Niro, the Garfield brothers themselves…

"My dad had this awesome old VHS on-the-shoulder camera that he was very proud of," explains Garfield when we meet on the Sony Pictures studio lot in Los Angeles. "We made a boxing movie once, which was really good fun. Dad was a good director, actually, very nurturing and good to his actors. The films usually ended up with me crying, though, because my brother accidently on purpose hit me too hard."

Even in his preteen years Garfield was showcasing both action chops and a ready ability to burst into tears, handy qualities for a Hollywood A-lister. "Movies were a massive part of my upbringing," Garfield continues. "My dad has always loved movies; he would drive past Fox or Paramount in Los Angeles when he was a young man and would dream about what was going on in there. He has this really romantic view of moviemaking. Now that I'm doing what I'm doing he is able live it alongside me a little bit."

Highlights for Richard Garfield this year will include watching his 30-year-old son produce his first movie, take the lead in a Martin Scorsese film – and, of course, save New York City once again, as he reprises his role as Spider-Man. His casting for the first instalment of the Amazing Spider-Man series, announced four years ago, took the industry and fans alike by surprise – Garfield had largely played troubled, sensitive souls, as in Robert Redford's Lions for Lambs, opposite Carey Mulligan in Never Let Me Go and, of course, as Mark Zuckerberg's betrayed friend Eduardo Saverin in The Social Network. Then again, Spiderman's previous incarnation was Tobey Maguire.

In the new film, Peter Parker sheds his teen angst and embraces his superpowers, battling a phalanx of villains played by Paul Giamatti, Jamie Foxx and Dane DeHaan. "I wasn't satisfied with the first movie," says Garfield, "but I think there is a cohesiveness on this film that we didn't quite have on the first. We were finding our feet a little bit on the first one." Still, the first film didn't do too badly – it grossed upwards of £450m in worldwide ticket sales, despite some clunky moments and a leading man who never seemed entirely comfortable when asked to promote the film. "I feel like I've changed a lot since I shot the first movie," Garfield says.

I've interviewed him several times in the past few years, and he's never been as excruciatingly uncomfortable as he was during the press tour for his first Spider-Man film, in 2012, where he mumbled his way through interviews, eyes cast to the ground.

Andrew Garfield, left, and Jesse Eisenberg in Columbia Pictures' The Social Network.

Today, by contrast, he is full of energy and charm. His tall, wiry frame, which is wrapped in black T-jeans and jumper, fidgets in the chair as he talks. He is a thoughtful chap – he quotes both Shakespeare and Cat Stevens during the course of our chat – and like Peter Parker in the new film, he seems more settled in his superhero skin. He makes regular eye contact and his rat-a-tat, many-words-a-minute delivery is punctuated by bouts of self-deprecation.

"There is something crass about self- promotion," he says. "I can't do the Twitter thing either. Maybe there is something wrong with me. Maybe I need to be better at saying: 'Check me the fuck out, motherfuckers.' I should do that." He laughs and deepens his voice, "Yeah, so go see the fucking movie," he says jabbing his index figure at an imaginary camera, "because it's the best fucking thing that you will ever see!"

Garfield was born in Los Angeles in 1983 and was raised in England, his parents moving to Epsom in Surrey when he was three years old – his mother, Lynn, is orginally from Essex. His father was born in the US and his paternal family name was originally Garfinkel – his grandfather, Samuel, switched it to its current incarnation.

"I enjoyed my childhood, though looking back on school, that was both great and awful," he says of his youth. "My best mates in life are my best mates from secondary school. There is something about being bored together every day for six or seven years. Your roots get entwined. But my sensitivity and my thin skin got in the way of me having really a good time in school. I am stupidly sensitive."

Andrew Garfield in Boy A (2207), for which he won a Bafta. Photograph: Allstar

It was a trait that attracted the attention of the school bully, who taunted him over his skinny frame.

"I didn't have this really abusive experience." he says. "I had the general bullying experience where there was a kid who wasn't very happy and he threw his weight around. He targeted me because I was skinny and thin-skinned and he knew it.

"It is really interesting, though," he continues, "what you perceive as your wounds can later on turn into the biggest gifts you have. At the time, I was like, 'Man I've got to toughen up, get a thicker skin,' whereas in fact, being sensitive and thin-skinned and all the things I still am, lends itself to a career as an actor. Also, if it weren't for that bully, I wouldn't have found Spider-Man. That's why I got so lost in the comics and the cartoon series. I would fantasise about being him, and I would fantasise about him coming to beat the crap out of my bully."

It might sound a handy line for promoting the movie, but Garfield's delivery is deadpan, and he did grow up a bona fide Spider-Man nut, famously dressing up as the red and blue web-slinger for his first Halloween party. His journey to the real costume began at the Royal Central School of Speech and Drama and he was turning heads soon after his graduation with stage performances in Kes and Romeo and Juliet, winning the Manchester Evening News award for Best Newcomer in 2004 before scooping Outstanding Newcomer at the Evening Standard theatre awards two years later.

He shone on TV, too: in 2007 he summoned a troubling performance in the television film Boy A, a story that shares similarities with the James Bulger case, winning a Bafta for his portrayal of a young man struggling to reintegrate into the social fabric in the aftermath of his childhood crime. And he brought a cocky recklessness to his newspaper reporter in the first chapter of the 2009 Red Riding trilogy.

Actor-director Robert Redford gave him his cinematic debut alongside Tom Cruise and Meryl Streep in Lions for Lambs, but his breakthrough came with The Social Network. He met the film's director, David Fincher, to discuss the role of Mark Zuckerberg, but Fincher cast him instead as Eduardo Saverin. "I was like: 'Why would we waste an actor like this on the part of the guy [who seems like he has] Asperger's?'" Fincher said at the time. "'I'm trying to cast somebody who doesn't have this access.' And yet that is Andrew's greatest strength."

Caught in his web: Andrew Garfield with his co-star and girlfriend Emma Stone. Photograph: Kevin Lynch

It was this emotional access that brought Garfield to the attention of the Spider-Man producers – who were looking to reboot their franchise following the termination of the series by Sam Raimi – and also to theatre director Mike Nichols, who was casting the 2012 Broadway revival of Death of a Salesman and was still searching for his Biff.

"What you could see in The Social Network is Andrew's enormous emotional equipment," Nichols said in the New York Times during the play's celebrated run. "In the scene where they'd stolen the company from him he was astonishing. He was practically exploding."

The actor's performance as Biff earned him a Tony Award nomination, proving, in case anyone had forgotten, that he is not just a movie star – Garfield is a wonderful stage actor, too. "I often think that if all this movie palaver fails and I never get to make a film again, I know that I can always do theatre," he says. "I could rent a studio apartment in London and I know I'd be happy. Making money has never been the main intention. Because my father was a businessman, I became allergic to that."

Indeed, for all his success Garfield has still not purchased a home. He lives an itinerant lifestyle, he says, with a Vespa and a surfboard in LA, a storage unit in New York and the majority of his personal effects at his parents' house in Surrey. When he is not working, he spends much of his time with his girlfriend and The Amazing Spider-Man co-star Emma Stone, who features in the series as his on-screen belle, Gwen Stacy.

"What I like about Andrew is his passion," Stone tells me when I speak to her later in the day. "He is completely able to transform and he looks at every facet of the character that he is playing. It looks like it might be a potentially exhausting process, but he is so invigorated by it. He is the most prepared actor I ever worked with. He leaves no stone unturned."

Ignoring the obvious pun, I wonder: what does Garfield adore in Stone? Their relationship has been hiding in plain sight ever since they met on set of the first Amazing Spider-Man film and they've both previously dodged questions on the subject whenever it has come up.

The moment the question leaves my lips, Garfield's smile suddenly drops and his eyebrows knit into a scowl. "What can I say? How about: 'Fuck off you fucking…'" Cue momentary alarm before, thankfully, his face relaxes and he laughs out loud. "No, I'm able to talk about her, but I want to protect something that I respect in my life," he says.

"I don't feel the need to speak publicly about my private life. And on a professional level I don't want people watching me on screen to know about my dirty laundry.

Andrew Garfield and Emma Stone at the world premiere of the first The Amazing Spider-Man in 2012. Photograph: Kento Nara

"It's not that interesting anyway. I'm about to do a film where I play a Portuguese priest and if people are reading about my personal life, they'll be watching the screen going: 'Oh that Portuguese priest is having sex with whoever!'"

The Portuguese priest will make his appearance in Silence, a forthcoming feature from Martin Scorsese, which goes into production this winter. Starring in a Scorsese film is a reminder that Garfield has much more to offer the world of Hollywood than shooting webbing from his wrists. The film is based on Shu-saku Endo-'s 1966 novel: "It's a wild one," says Garfield. "I play a Jesuit priest travelling to Japan in the 1600s."

He calls it a father-son story (the father, in this case, being God), and the same theme emerges in 99 Homes, the first film he has produced himself, in which he plays a single father battling eviction from his Florida home. "I thought that the story was fascinating," he says of his decision to invest in the film, "because I was raised with a value system that emphasises community and family."

Does he dream of starting his own family one day? "Fatherhood is a scary thing because as soon as you've done it you've messed up the kid," he says. "You're only human and you'll be projecting your own shit on them and putting your own expectations on them and treating them as though they're a part of you rather than a separate entity.

"I am absolutely fascinated by fatherhood, though, and have been for a long time," he adds. Presumably, when the time comes he will be out there with his camera, making movies with his kids just as his father did with him? "I hope so," he says. "I intend to be the perfect father. I'm sure I'll fail miserably."

The Amazing Spider-Man 2 will be released in the UK on Wednesday 16 April