As others have said, most of the stuff you've been watching is just...It's positively drivel (In my opinion, of course). Yu-Gi-Oh, Pokemon, and shows like it are literally just 20 minute commercials. The show was made for the sole purpose of selling a separate product, with telling a meaningful story being an afterthought. They're essentially Japan's version of G.I.Joe.
By openly asking for recommendations, you'll be getting them up the ying-yang like crazy, probably overwhelming you with choices. And I'm not going to help with that, in fact, I'm gonna make it worse.
Animes that I think are worth watching:
Black Lagoon
Already been recommended by most here, but I must repeat it. This is an excellent show, and I highly recommend watching the first couple episodes. Get the English dub if you can, it was really well done.
Spice & Wolf
A very obscure show, that is done in almost a western fashion. No inappropriate goofy-ness, no fan service, just an incredibly interesting story, with very realistic characters. Also has a stunningly good dub.
Now, on to the premise. It's about a traveling merchant in a medieval world, who encounters a strange girl. With smatterings of medieval economics, action, and the best romance I've ever seen in any medium EVER. I of course recommend it fully. It has a slowish start, so give it at least until episode 3 before you give your final opinion.
First episode, if you're intrigued:
Steamboy
Anime movie made with incredible production value, about a boy inventor in a steampunk world. Excellent movie, and it even has Patrick Stewart in the english dub!
Trailer, if you're interested:
Sword Art Online
Pretty decent anime with a romance tone...Just don't watch the second season, and pretend it ended with the first one.
Attack on Titan
Also a decent show, but it does have some filler episodes.
So...Yeah, diversify your anime viewing habits.
Also, you say you've seen the bebop movie, have you seen the TV series as well?