Depending on your favorite genre of movies, I'd recommend.
Love story = Get any of the Tenchi Muyo series, Ah! My Goddess, or Love Hina.
Very cute and sweet, Tenchi and Love Hina have lots of comedy with the romance, and Ah! My Goddess is just a great romance, and it takes a lot for me to say that.
Action/gunfighting = Cowboy Bebop, Trigun. Welldone, with the added great music, and indepth stories.
Psychological "Makes you think movies" = Serial Experiments Lain. Best when watched when sleep deprived. A girl discovers the matrix (called the Wired in the show), becomes addicted and actually splits her personality. Her digital form is actually acting indepently of herself (In the sense that there are two separate identical people. Not just two in one head.). A good study of how people freak out when reality starts to bend. Good for wanna-be deckers.
You must see Neon-Genesis Evangelion. It's obstensibly a mecha anime, but it's got deep, deep symbology about family, religion, man vs. god in a war, social pressures, etc. In fact Serial Experiments Lain is like that too. I am a fan of the "Very deep symbolic, figure it out" type movies.
P.S. FLCL, also called Fooly Cooly, aka Furi Kuri is a great show to watch! It is only six episodes long. But it is, how should I say it. On the surface it's a wacky, zany show full of funny moments and great action. Then underneath that is the huge depth that I've previously discussed. . . Made by the same people who made Evangelion.