Princess Mononoke- The plot is good, dubbing work is excellent, and it's worth watching quite a few times over. It's what sold me on anime in the first place.
Outlaw Star- An excellent space western. Granted, it's no Firefly, but this was so good it helped inspire Whedon to make Firefly in the first place.
Excel Saga- This comedy series is the most batshit crazy thing you'll see. I guarantee it. It's also a parody of nearly everything.
Lucky Star- It's like anime Seinfeld... with high school girls. It's a show about nothing, but manages to be hilarious.
The Melancholy of Haruhi Suzumiya- This popular series continues to meet the hype around it. The premise is, basically, a Japanese schoolgirl is God, but doesn't know it. If she gets bored with the world, she'll end it and make a new one. People thus try to keep her highly ADD self entertained, and hilarity ensues.
EDIT: Also, all these that I recommended (except Princess Mononoke, which is a 134 minute movie) are only one season long, so it won't take you forever and a day to watch them all.
EDIT AGAIN: Though I'd love to see more seasons of all of them released over here.