I'm not a big fan of anime - mostly because I can't get past the writing, I think - but there are a couple that I love.
Dominic's right, start with
Akira. This is the first anime that made a big crossover into the US market.
Second, and my personal favourite,
Ghost in the Shell. Despite being 15 years old now, it still has some of the best animation you'll see anywhere, and even if the dialogue is clumsy, the story isn't. I wish James Cameron had done something with the option to do a live action version he had for a while. There is no love story, despite the heroine being the hottest thing since Jessica Rabbit. Forget the sequel and the tv series.
Third,
Grave of the Fireflies. Unlike the other two, which are action films, this is a drama. It's about two children trying to survive during WWII in a bombed out Japanese city. Don't expect to have fun, do expect to see one of the saddest films of any sort you've subjected yourself to.
Additional suggestions if you don't want to go with the drama: Ninja Scroll, Perfect Blue, and Metropolis (this is an anime version of the Fritz Lang film).