For a young person new to anime I would suggest a series that isn't too serious, but has plenty of action and comedy. Something like Naruto (of course), Trigun, or One Piece.

From there you can see what other genres you'd be interested in. My favorites are the pure action movies (Ninja Scroll, Spriggan, Vampire Hunter D) and sports anime (Prince of Tennis, Hajime no Ippo, and I'm gonna try to fit Initial D in here).

If you're older, it would be better to start with a genre that you already know that you like. If you like comedies, you could try Goldenboy (18+) or Excel Saga(18+ for last episode). Most adults who are new to anime find something that they like in Cowboy Bebop, but not everybody.

What are you interested in?