Steer clear of crap like Full Metal Alchemist, Naruto or Death Note. With only a few examples, most anime you’ll find on American networks is watered down to comply with censors.
But anyways,
I’m partial to Cowboy Bebop because of it’s noir storyline and heavy use of jazz and blues in it’s soundtrack. Since I’m a music junkie I think it meshes well, but you can decide for yourself.
Also, try Hellsing, it’s pretty interesting for the whole concept, and I liked it a lot.
Ghost in the Shell, both the movies and show are also fairly interesting, but I felt that the plot got old after a bit. Inspired a bit by Asimov, with heavy emphasis on robots and humanity. I love it for the concept, but the cop angle of the show is a bit tired.
The exceptions to programming on American networks that I think are good usually air late night are few and far between, but Samurai Champloo is worth checking out. Again, the soundtrack in this is more traditional and modern downtempo Japanese theme; mellow hip hop or traditional instrumentation. And the animation is pretty nice too. Neon Genesis Evangelion used to air too, though I’m not sure if it still does. Very dark, in my opinion, deals a lot with the human psyche and topics of insanity, religion, creation, oppression, things like that. Follows true to regular style in being really weird and employing mecha-based action. Also, Fooly Cooly (pronounced as “Furi Kuri”) is particularly interesting. Basically, a major animation studio—I think the one responsible for Neon Genesis Evangelion—took a break for a while on their hit and tried to produce a wreck of TV show, doing things they normally couldn’t have on their hit. So the show is really weird and has some awesome music… and more importantly makes little to no sense.
As far as movies go, I’d strongly recommend Akira as a starting point. It’s a classic anime, inspiring lots of other concepts. In a similar vein, try Metropolis. The Girl Who Leapt Through Time and Howl’s Moving Castle are cool too. After watching those, try Appleseed Ex Machina, Iblard Jikan, or Gin’ro no Kami no Agito (hopefully you don’t mind subtitles). I just watched Steamboy too, which is a bit new but I loved every minute of it.
Oh, and speaking of subtitles. Unless you detest reading (hopefully not, it’s good for you!) I always recommend using the original audio—usually in Japanese; again, steer clear of Americanized “4Kids” programming and et cetera—with english subtitles. If you’re downloading this on the internet, it should be easy enough to obtain. If not, most DVDs usually ship with original and subs. My reasoning is that even good dubbing often doesn’t stay true to the original character’s voicing; i.e., a character that sounds tough or mean might just sound pompous in English. There is a certain tone that is hard to translate across languages.
So in closing, here is an ordered list of series or movies to check out (ordered from best, in my opinion, to worst):
Cowboy Bebop
Neon Genesis Evangelion
Fooly Cooly
Samurai Champloo
Hellsing
Ghost in the Shell
Akira
Metropolis
Howl’s Moving Castle
Steamboy
Gin’ro no Kami no Agito
The Girl Who Leapt Through Time
Iblard Jikan
Appleseed Ex Machina