Recommend me some anime!

Are you looking for good new stuff, or just good stuff? What kind of stuff are you interested in?

Not new stuff that's good:
Trigun and Cowboy Bebop are both great action shows, set in a frontier-in-space setting.
Onegai Teacher is about the romance of a student who married his teacher (who is an alien) to get out of trouble, but they fall in love later.
.hack//SIGN is a drama about a boy who mysteriously finds himself trapped in an MMORPG. Is he a victim or a villan? Is this indicitave of something bigger happening? Does he really want to log out?
Noir is an action/drama about two assasins fighting to find the truth about their pasts.
All of these are excellent and will probably hold your interest even if you aren't really into their genres. All are available on region 1 DVD

New stuff that's good:
Simoun is a drama about priesteses that are forced to become soldiers, their fight with their feelings, their fight to become one, and their fight with the truth. This series finished in the second half of 2006. The full series is available subbed, if you look hard enough. Simoun-Fans is currently finishing up their sub of it.
The Melancholy of Haruhi Suzumiya is a comedy about a girl who is omnipotent and doesn't know it, and those she surrounds herself with who do know and are trying to prevent the end of the world. I believe this series finished in 2005. It is available subbed. It has been recently licenced.
Ouran High School Host Club is a romance/comedy about a poor girl who is going to a rich high school and finds herself playing the part of a male "host", which is a gentleman who entertains women over tea and sweets. This show is hard to explain, but is great... I believe this series finished in 2005. It is available subbed. The manga is available in part in the US.
Soukou no Strain is a mech action/drama about a girl who's brother turned sides in the war and took everything from her. This show is still airing and is available subbed up to episode 9 so far, with dual subbing groups.
Kanon is a a romance based on the VN of the same name. Yuichi returns to a town that he visited for a time seven years in the past; his memory of that time is gone, but the town's memory of him is not... There is an earlier version, but the current one is much better. This show is still airing and is available subbed up to episode 14 so far, with dual subbing groups.
Busou Renkin is an action show about a boy whose life was saved by Arms Alchemy, who picks up the fight against Homonculus, creatures created by alchemy who eat humans. This show is still airing and is availabled subbed up to episode 13.