Several good recommendations in this thread, but I'm sad to see Escaflowne left out of so many lists. I never watched this classic show, and I just finished the series for the first time. I wrote a silly little thing for it too over in the Anime thread, so check it out of you want to know my feelings about it.
For beginners I also recommend:
Cowboy Bebop
Code Geass
Gurren Lagann
Baccano!
Samurai Champloo
Gunbuster
JoJo's Bizarre Adventure 2012 (currently airing)
Those are all easy entry points, and should prove to be highly enjoyable and very accessible. After that I'd recommend Dennou Coil and Spice & Wolf.
I can agree with pretty much everything here, assuming you approach Code Geass as a (mostly) inadvertent comedy. (This goes for Death Note on any lists that recommend it, too.)
All these are quite good, I especially love the first two.
Monster
Legend of Galactic Heroes
Death Note
Serei no Moribito
Gungrave
Berserk
Planetes
Samurai Champloo
Escaflowne
Paranoia Agent
FMA Brotherhood
Trigun
This list is even better, aside from its lack of Bebop. Pretty much everything here is top shelf. When Gungrave and Death Note are probably the worst shows on the list, it's a pretty solid list.