When it comes to youth, Hollywood is like a game of tag. One minute you're zipping around the playground with all the other kids. The next minute you're It. Your name rolls off the tongue of everyone from Sharon Stone to Francis Ford Coppola. Your face beams from the pages of magazines. Deep down, you're still a kid, but from now on your every move will determine the rules of the game for the rest of your life. Such is the state of affairs for Leonardo DiCaprio, a prodigiously talented performer who received a Supporting Actor Oscar nomination for 1993's What's Eating Gilbert Grape? With River Phoenix gone, and with current hunks like Keanu Reeves, Johnny Depp, and Brad Pitt maturing, the 20-year-old DiCaprio, who shoots heroin and hoops in April's The Basketball Diaries, is, as they say, a shooting star. Oddly, DiCaprio has become the great new hope even though not one of his films-Grape, Dead, and 1993's This Boy's Life-was a hit. Instead of box office tallies, it's DiCaprio's blazing talent and dashing baby-faced looks-a combination of the mystic and the mischievous-that have the praise faucets gushing buckets: ''Leo embodies the spirit of youth,'' declares his Quick and the Dead director Sam Raimi, who cast him as an Oedipal gunslinger who duels with his dad (Gene Hackman). ''Fabulously cool,'' says Diaries producer Liz Heller. ''He's just centered and still so young, with that energy.'' ''He's so good, it's scary,'' says Stone, who sparred with DiCaprio in Quick. ''I was dying to have him be in this movie. I would have carried the boy on my back to the set every day if that's what it would have taken. Luckily, Leonardo is down-to-earth and walked by himself.'' As if a piggyback from Stone weren't enough, Coppola calls DiCaprio ''gifted'' and wants him to star as Jack Kerouac in his silver-screen version of On the Road. ''He has a natural talent or ability to slip inside a role,'' says Coppola. ''Plus, he has a sweetness and attractiveness.'' Meanwhile, folks at Warner Bros. have been hankering for DiCaprio to play yet another rebel icon, James Dean, but so far that big-budget biopic has been foundering: Original director Michael Mann (The Last of the Mohicans) bowed out, and Broadway's Tommy whiz Des McAnuff followed suit. The flick's status, says a Warner spokesman, ''is not ready to be discussed in any serious way.'' Use him or lose him. DiCaprio has reportedly been approached to costar opposite Richard Gere in Paramount's upcoming murder mystery, Primal Fear. But even in a town that thrives on speed, DiCaprio's rise-starting when the then- 14-year-old Los Angeleno appeared in an educational film called How to Deal With a Parent Who Takes Drugs-has been remarkably, well, speedy. After mugging in the television sitcom Growing Pains, he landed a role as Robert De Niro's abused stepson in This Boy's Life, and left the boob tube behind. ''Funny, but I thought that on my gravestone they were going to write, 'This was the guy from Growing Pains,''' DiCaprio once said during a publicity tour for This Boy's Life. ''Please don't write that I was on that show. It's sort of embarrassing.'' Having played one junkie poet in Diaries, DiCaprio is currently in Africa wrapping up his role as another-the 19th-century French enfant terrible Arthur Rimbaud-in Fine Line's Total Eclipse. Despite rumors of DiCaprio's putting too much Method in his acting, he seems to be handling his growing pains with the grace of an old pro. ''There's a lot of pressure being the Next Big Thing,'' says Marcia Ross, now a vice president of casting at Disney, who first spotted DiCaprio while trolling for talent at Warner Bros. TV. ''It's a lot of pressure to have people watching you.'' She believes DiCaprio should try everything, including ''a flat-out comedy. When you're young, people can't label you. The great actors can do lots of roles.'' New Line, meanwhile, intends to transform DiCaprio from critical darling to American stud muffin. ''On Basketball Diaries our job is to get beyond the art- house circuit and reach young people in the neighborhoods,'' says the studio's marketing president Mitch Goldman. ''I don't know what Leonardo's name means out there. All I know is, if you don't remember him from Gilbert Grape, you're going to remember him after this.''