Uh, not sure I agree with you on the visuals part. Makoto Shinkai always goes for overkill when it goes for visuals, and while they aren't 'artistic' as much as say Spirited Away, I'd rate the art on this above pretty much everything else, definitely significantly higher than Bebop.
At Berka: There's a lot of great anime content out there, personally I wasn't that much of a Bebop fan. I'd recommend (in rough order of preference) FMA: Brotherhood (head and shoulders above everything else on this list), Code Geass (the ending is a major payoff, both seasons), Steins;Gate, Clannad with After Story (the first season is fine, the After Story dials it all beyond 11), Psycho-Pass (first season only), Planetes, Shirobako, Hunter X Hunter (2011), Ghost in the Shell: SAC, Gurren Lagann (goes from a 7 to an 11 the moment Season 2 starts), No Game No Life, Attack on Titan, Spice and Wolf, Kill la Kill (pleasantly surprised by this one), One Punch Man. Note, that list contains a large variety of genres, so there's something for everyone.
In case you haven't seen it, Avatar: The Last Airbender is a paragon amongst Western animation, followed by a worthy but not outdoing follow series The Legend of Korra. For something more mature, Bojack Horseman is excellent, as is Rick and Morty.