For no particular reason, and without claiming to be an expert at all, I thought I'd list some anime series I like. In no particular order:
Paranoia Agent. Coherent weirdness by the late master director Satoshi Kon.
Cowboy Bebop. Very stylish science fiction noir. Great characters. The movie is great too.
Samurai Champloo. Very stylish samurai action by the director of Cowboy Bebop. Flash forwards into the current day? Beatboxing samurai? Check. Amazing fight scenes.
Ghost In The Shell: Stand Alone Complex (aka GitS:SAC). More like the second movie Both series are excellent. There is a SAC movie and a new series, all good.
The Melancholy of Haruhi Suzumiya. A story about an annoying high school girl. Saying more, including the genre, would be spoiling things. The episode order has been randomized because why not tie one hand behind your back? Touching, hilarious, exciting, coherent.
Serial Experiments Lain. Twin Peaks meets 90s cyberpunk.
That's it. I've seen other series, but… I didn't like them as much as these ones.