I'd go with what the majority of people are saying: Death Note's a fantastic place to start. Once I started watching, I was totally hooked
Definitely go for a bit of bebop- fantastically well-done, and also great to watch dubbed if you dont want to read subtitles too much.
those two are quite adult titles.
If you're into vampires, hellsing's a good watch, but it can run out of steam a bit at points.
full metal alchemist is a bit slow for my taste, but it's got some bits which are fantastic.
It depends what type of films you watch really:
if you like thrillers, go for death note, or full metal alchemist for something a little less twisty.
if you like sci fi, definitely go for bebop, or ghost in the shell.
if you're looking for something light-hearted (but not 'cute') go for one piece or naruto.
if you're looking for something more resembling a romcom, i loved chobits.
if you like the idea of overpowered things beating up other things, go for hellsing for a short burst, bleach for a longer burst.