Good starting points:
Neon Genesis Evangelion+ The End of Evangelion
Death Note
Cowboy Bebop
Space Dandy (currently airing, on toonami I think)
Angel Beats
Code Geass
Gurren Lagann
Steins;Gate
Trigun
Fullmetal Alchemist Brotherhood
Movies:
Akira
Tokyo Godfathers
Perfect Blue
Redline
Jin-Roh: The Wolf Brigade
Tekkon Kinkreet
The Girl Who Leapt Through Time
Ghost in the Shell
5 Centimeters Per Second
There's some good/popular entry-level stuff. Evangelion is what got me into anime and it's still my favorite, but it isn't for everyone. Death Note seems to be the most popular entry point and it is pretty great for the most part, I'd recommend starting with it. You'll probably figure out where to go from there.
Don't bother with the super huge shounen series (Naruto, One Piece, Bleach, DBZ) as they're shit unless you grew up with them and they have so many episodes it's ridiculous. Don't bother with Elfen Lied.