Well, let's see. Most of my suggestions are going to go towards the weird and haunted shores at world's end weird and super crazy, plus bloody and awesome.
Higurashi no Naku Koro ni: It's about a town named Hinamizawa and the inhabitants who live there. Definitely a good watch, and don't worry about what happens in the beginning; keep watching. There are two seasons and an OVA. Western name: When They Cry (though the technical translation is 'When the Cicadas Cry'.
School Days: This anime is completely malicious in how tame it is in the beginning. Worth the watch just to see how badly it goes at the end. There's the series and two OVAs.
Umineko no Naku Koro ni: Made by the same people as Higurashi, but featuring an island and the same recurring time loop (a la Higurashi). Not quite as good, but I liked it and it's worth a watch. Western name: When the Seagulls Cry. Just one series so far.
Ga Rei: I'm beginning to think I have a thing for crazy women; oh, wait, I totally do. This one is about two girls who are part of different family lines that can summon monsters from tattoos on their back to fight evil spirits. The anime is a prequel to the manga, and both are worth going for.
Akira: A must-see for any burgeoning anime fan. This is the prototypical anime that really got things going in the US, I think. It's about mental powers, friendship and gang-riddled Tokyo (I think it was Tokyo). The motorcycle is fucking sweet, too. This is a movie, I'm not sure i there's a manga or not.
Kaze no Tani no Nausicaa: Nausicaa of the Valley of the Wind. One of my all-time favorite movies, by director Hayao Miyazaki. I'll go ahead and say that any movie he's done is worth watching. Spirited Away, Howl's Moving Castle, Ponyo, Mononoke Hime, he even did one of my other favorite movies: Lupin III and the Castle of Cagliostro. (Lupin is a series of movies you should definitely check out when you have time).
Instead of continuing to list all of the anime I'd recommend, I'll just drop this here: http://anidb.net/perl-bin/animedb.pl?show=mylist&do.filter=1&uid=245534
This is my aniDB list. Ignore the DBZ on there, I watched it before I knew any better. Anything else, give it a watch, they're all good. Some are NSFW, be warned.