This list is like noob anime central. It's basically like saying, "Do you like jazz? I love jazz! Miles Davis is my favorite!"
(I'm not sure that I actually like anime. I just like anime auteurs like Shinichiro Watanabe, Hayao Miyazaki, Satoshi Kon, etc.)
tl;dr: Watch Cowboy Bebop/Samurai Champloo for a series and anything Miyazaki or Grave of the Fireflies for a movie.
- Cowboy Bebop (One of the greatest things ever, although the movie was meh. Also one of the only ones I can safely reocmmend in English.)
- Samurai Champloo
- Kids on the Slope (I generally don't like "slice-of-life," but I love jazz and Yoko Kanno and Shinichiro Watanabe, so I loved this.)