All anime recommendations:
Perfect Blue -
very creepy film; deals with a falling pop star and the people dying all around her.
Spirited Away - Hayao Miyazaki's masterpiece as far as I'm concerned.
My Neighbor Totoro - it may be a family film, but Miyazaki can make them absolutely amazing.
The Grave of the Fireflies - feeling too happy lately? Then watch this film and be utterly depressed.

(Seriously, this is a fantastic film about two Japanese orphans during WWII.)
Ghost in the Shell - watch the new edition on DVD (the one with the silver cover); it has a better dub, which helps in understanding this cyberpunk classic.
Cowboy Bebop: The Movie - just for the hell of it; can't go wrong with interplanetary bounty hunters.