Listen to roujin, most of his list is spot on. Definitely check out: Trigun, Cowboy Bebop, Berserk, Beck, Yakitate Japan!!, and GTO. I myself need to see Gankutsuou, Cromartie HS, and Monster.
Lupin III: The Castle of Cagliostro
It's an anime heist movie that's easily the best of it's kind.
Tengen Toppa Gurren Lagann
features a pretty epic (not the lame internet meaning) story and ending.
Jungle wa Itsumo Hare nochi Guu
Is a light heart childish comedy, but it's probably the funniest anime to date and one of my favorite anime.
It's another classic action anime that's similiar to Trigun. It's up there with Trigun and Cowboy Bebop as the best.
It's a realistic outer space anime. I myself haven't seen this yet, but it's supposed to be great.
Gunbuster and Diebuster (Gunbuster 2)
Great short series with phenomenal endings that I dare you not to cry at.
This one's all over the place, it has gangsters and plenty of action.
About a gambler down his luck trying to erase the debt he's in. The main character gets more badass as it progresses.
This is sort of like a Pulp Fiction of anime. It follows multiple people and connects them together in one story. It has the coolest couple of any anime I've ever seen.