I'll add my own list of suggestions, because I think I've got one that no one else seems to have noted.
*Note: The following list is only anime that I have watched at least half of. I'm not the biggest otaku, but I'm pretty confident I can tell a good anime from bad*
Dragonball/DBZ - It's been a while since I've watched it, but I enjoyed it. The best pre-Fruits Basket dub, in my opinion. If you do watch, only go up to the Cell Saga, for your own sake. It just gets ridiculous from there.
Death Note - Just amazing. The constant chess match between Light and L kept me guessing for a long time. I know I'll never see anything like this again.
Azumanga Daioh - A good, light-hearted anime. Pretty funny. Please don't watch the dub.
Lucky Star - See Azumanga Daioh. They're pretty similar in style.
Love Hina - Just plain bizarre. I loved every second of it.
Negima - If you like Love Hina, this is by the same writer. The series makes a dramatic shift from comedic to serious about two-thirds of the way in, though, which caught me off guard.
The Melancholy of Haruhi Suzumiya - Though it seems like pure comedy at first, this anime can get pretty deep. Makes you think about a lot of stuff. The order in which the episodes are shown is perfect, as well.
Black Lagoon - This series is action-packed and has pretty interesting characters, although it can be a bit somber and strange at times. Just to give you an idea, the scene that got me hooked was when a helicopter pilot took a torpedo to the face in the second episode.
Full Metal Alchemist - The canon behind FMA is incredibly detailed, and all makes sense once you get your head around it. The characters are also easy to become attached to, so when one of them gets killed off, it's pretty heart-wrenching. One of the few fantastic English dubs out there, as well.
Finally, my favorite anime:
CardCaptor Sakura
No, I'm not kidding. My favorite anime (A 20-year-old male college student) is a shoujou about two 10-year-old magic users.
There's a lot to this series that makes it great in my mind. The characters are captivating, the story kept me entertained, and each episode shows a little change in each character, which is capped by probably the most touching ending scene in an anime I've ever seen.
If you do watch it, do not watch the dub. Multiple characters and scenes are completely ruined because of American "censorship".
So, there you are. My list of recommendations. Enjoy your future endeavors into the world of anime.
