Casual Shinji said:
Katsuhiro Otomo's Memories
Seriously, track this movie down. It is pure awesome sauce.
Man, everything Katsuhiro Otomo touches is a fucking masterpiece. Memories was amazing, Akira (needless to say is an absolute classic), and I adore the live action adaptation he did of Mushishi.
As for animes I would suggest. Well, as a general rule I kind of like stuff that really makes you think, series with deeper philosophical, theological, and metaphysical themes really appeal to me, so keep that in mind when listening to my suggestions.
Anyways, Kino's Journey and Mushishi are two of my absolute favorites, they tend more to the quiet contemplative side than being heavy on action, which I like, so if you want something thoughtful and calm these two are excellent.
I just finished re-watching Ergo Proxy for the second time. It actually didn't really hit me just how good it was the first time through, but the second time through really sold me on it. At first it kind of seemed like another tired plot that I'd seen played out a million times before in the general futuristic utopia sci-fi genre... but if you'll give the show ample time you'll find that it's much more sophisticated than that, tending less to rehashing 1984 and more to attempting to understand the relationship between a hypothetical god and its creations.
Two other classics you might consider are Serial Experiments Lain and Neon Genesis Evangelion... I mention them together because even though they're extremely different, they do have some thematic similarities, and their style of presentation is pretty similar as well. Both of them will mess with your head, Serial Experiments Lain in particular. I would also say that both of these, especially Lain, are shows which can not be fairly judged by just one episode, so if you're going to watch them then set your resolve so that you watch them in their entirety. They're artistic masterpieces too big to be understood either as a whole or in their pieces, but if you only see the pieces then there's no way you'll be able to understand just how good they are.
Finally, I would suggest Black Lagoon. It's got some pretty badass action, great characters, and especially near the end, gets into some pretty heavy existentialist themes... and by pretty heavy existentialist themes I mean that they're quoting Sartre... By the time everything is said and done it gets pretty dark, but it's really good too. It's also a little easier to get into than Lain or Evangelion, as you could get a general appreciation for it just from watching a couple episodes, though, again, to really understand what it's saying it's best to watch the whole thing.