Cowboy Bebop, it's really my first anime (if I don't include Pokémon, my awareness of anime was zerozerozero when I was 10 or so). And I loved every second of it. You should just watch it for the amazing music, and then realise: wow, it's just a great series.
The story: Well, it is about a handsome man-in-suit, Spike Spiegel, who has to settle something with his former-friend Vicious; and of course, it's about a woman. But don't think: Aw, again one of those shallow crappy animes! Did I tell you it is set in the future? I never really liked all those machinie-space-let'sliveonotherplanets-thingies. Cowboy Bebop is somewhat different, although I can't explain it very well.
The art: This is not the best drawn anime ever, I can name 10 other anime whose art is better and such, but it fits the series. The universe-stuff with all the flying vehicles it pretty realistic and the fights are nice.
The music: Aahh, the music. Yoko Kanno is a real music genius. There are sad melodies, but also very funky, jazzy themes. Especially jazz.
The characters: You get every one of them. Spike, the cool, laid-back man, who'll do everything for the woman he loves.'
Jet, the caring, robotarm, former-policeman.
Faye, the pretty woman with a huge debt, oh yeah, she's howmanyyears?
Edward Wong Hau Pepelu Tivrusky the fourth, a weird, young, female computer hacking genius.
And last but not least, Ein! The cute data-dog n____n
Are you still reading?
COME ON; WATCH IT?
See you space, cowboy :3