Cowboy Bebop: If you haven't watched it, watch it. It's an anime that is deliberately aimed at a western audience and it
nails it. The beginning is charming, the journey is heartfelt, and the ending is surprisingly touching, for a cartoon.
"See you later, Space Cowboy."
Fullmetal Alchemist: I'm sure others have recommended it, but they do with good reason. Brotherhood is darker than the original take on it, but both are incredible well done in crafting the tale of two brothers who stumble on far more about humanity than they would have originally imagined.
Naruto: Hahaha, no. Believe it.
Blue Gender: A little known anime with a rather B-Grade plot that gets the job done. It features mechanized troops fighting to retake Earth, with a select few infected with a virus that allows them to inherently understand the enemy. What looks like a rather mediocre sci-fi anime turns out to be a stark look at man versus self versus nature, and features an extremely well done romance between the two main characters, along with interesting side characters. Spoiler alert: Some of them die. Good luck guessing who, it threw me for a couple loops when I watched it the first time.
Neon Genesis Evangelion: If you like mindfuck stuff, this will rock your socks off. If you don't, avoid it. Aesthetically though it has some rather interesting choices. Something to keep in mind while watching: The screenwriter was facing suicidal thoughts when he created the original script, and each character is a manifestation of his own trouble mind. When you view it in this context, it goes from "what the fuck?" to "oh, oh fuck, I'm viewing the inner mind of a man brought to the edge of
madness and suicide."
Eureka Seven: This anime is all sorts of cute. It's got your slice of life... ON A SHIP! It has surfboarding... ON AIR! It has a cutesy prepubescent couple that draws you in and makes you fall in love with it... And then... It hits you. I can't even say how without spoiling it, but needless to say, I came for the adorable sci-fi, I stayed for the drama and to see if they could hang on to life and each other.
There you go. Have fun.
EDIT
Also, another point for Blue Gender.
Some pretty kickass music.