There are so many possibilities with anime! There is long anime, there is short anime. There is sappy anime, there is violent anime. But here are my personal recommendations:
(I won't go into what each is about, because I'm just that lazy :P)
Sailor Moon: What started it all for me. Looking back right now, the animation wasn't all that spectacular, but it is action packed all the way through all 200 episodes. =3
Cardcaptor Sakura: Another magic girl anime. Esoteric, and mad funny. Consists of two season, about 50 episodes i want to say. Not all that bad :p
Revolutionary Girl Utena: I think you'll love this one Rik. It has action, comedy and a lot of obscurity when you really read into it. ;) 69 episodes long, but every one of them RAWKS!!
Neon Genesis Evangelion: Possibly the best anime you'll ever encounter. (If you're into this kind of story haha). 26 episodes I think and a movie which serves as the real ending.
Ah! My Goddess: Whether you watch the OAV, or the two year old anime, you'll have plenty of fun!! =3
Chobits: Cute cute.
and of course... the anime which does not fit into a single category:
Elfen Lied: there is a little bit of EVERYTHING in this anime. Horror, suspense, action, violence, gore, comedy, cuteness, drama, love. Only 12 episodes long, but OMG.