If you're just starting out, I say go with Trigun. Cowboy Bebop is insanely good, but save that for later. Trigun is hilarious, entertaining, and can still be serious. It's a good place to start out, I think.

Also, if you can, try watching Adult Swim once in awhile. They have a few different anime shows, like FLCL, Paranoia Agent, Samurai Champloo, Fullmetal Alchemist, etc that are good, and chances are you'll find something you like.