Hmm its a bit hard as my taste in anime might not match up with yours (I have some weird tastes). Loved Cowboy Bebop so will give it a go:
Paranoia Agent is a great mindfuck, but its not for everyone, I tried to introduce it to some of my friends who love anime and it didn't go down so well for them. If all you're looking for action then I would recommend that you avoid it, if a modern day setting isn't what you want then avoid it. If you want to experience a real mindtrip then yes.
Outlaw Star, very much like Cowboy Bebop but more over the top and comedic, you should have no trouble liking this.
Ghost in the Shell, I'll admit I'm not that big a fan of it. All of its deep psychological delvings get undermined by the gratuitous clothing (or lack of) of the main character, and I found the series to be pretentious and up itself, though that might be because I found the main character to be pretentious and up herself. It's immensely popular though and you'll probably like it, I liked the ones that did not feature the major much.
Ergo Proxy, a series you'll either love or love to hate it seems. I love it, but I can see why many others don't. Don't go into the series expecting a cyberpunk story, despite its appearance it very rapidly leaves its cyberpunk setting behind and enters into the realms of fantasy after a few episodes, hell it even tricks you over who the main character is. The individual episodes vary in quality along the way, I was a bit put off around I think episodes 7-9 I think but then it starts to pick up again for me.
Evangelion, love it but unless you're a fan of both mecha and freudian mindfucks then you might not. Starts off mecha, and progressively 'disintegrates' as the characters' minds do, though that was more due to budget constraints rather than artistic choice, and the ending (both of them) are one of the more controversial ones amongst its fans.
Fullmetal Alchemist, brilliant series (just don't watch the film). Set in a fantasy world that combines late nineteenth/early twentieth century technology and society with alchemy, which is portrayed and utilised by the characters in a very scientific way. Funny but also serious, unless you don't like fantasy or the setting I can't see anyone not liking it.

i'd say the same as some people before me - watch Samurai Champloo. i prefer it more than Bebop.
you might want to also try Ghost In the Shell: Stand Alone Complex (i'd recommend only the first series, the second is pretty awful).

I think that, first of all, you should watch Akira.
Hellsing is great, Full Metal Alchemist and Ergo Proxy too.
And Death Note.

I gave up on Bleach's manga around 2 and a half years ago, if that helps.

Though you should really, really, really watch the *new* remake-ish one, not the old series. The new series follows the manga throughout, which makes it a lot better (especially in the end, where the story of the original got kind of confused) and is of much higher quality in general.
it's kind of meh, though it managed to amused me for a while. Just start watching it and stop when you get bored. It's pretty fun initially. But even the manga has completely degenerated into fight after fight after fight after fight by now (unlike the naruto manga, which is still pretty good, actually).

If you want to watch a movie, watch Steamboy. I really enjoyed it (bought it for little money for my PSP as UMD).
Else i could second Fullmetal alchemist aaand The Last Exile. Both steampunk animes with great stories.

A couple of old but good anime series I can recommend would be "Lupin the 3rd" and "City Hunter": they have a tendency to alternate between silly and heartwrenching drama (ok, maybe I'm axaggerating a little), but are the first two anime series that I ever watched regularly.
I still don't watch much anime, but a few series that I watched and liked are "Wolf's Rain" (bit depressing) and "Excel Saga", which could be summed up as a series that parodies anime, mangas, japanese society, western animation, westernisation, literature, movies, and itself: it has to be watched to understand, really. :p

Anime: Hellsing OVA
Manga: Berserk
Anime Movie: Secret of the Sword + all Ghibli stuff

If you want something completely surreal, try watching Fuli Culi, it's quite an experience of its own !

I saw clips of Hellsing on YouTube. It looks messed up.

Also, a Bleach manga fight consists of one attack taking up all panels on a single page and then the next five pages being everyone reacting to that attack. And recently it has devolved into the author trying to make every attack/power even more "awesome" than the last. It reads more like asspull upon asspull. It's like following a massive freeway car crash, you know you should stop watching, but you keep looking just to see how bad it really is. Bleach definitely is the worst of the big three.

There are two versions. The OVAs are the DVD-version. They are close to the manga story line, and yes...very messed up.

Anything Ghost in the Shell. In video that is. Man Machine interface manga just feels like a techical manual with bare bollocks. on like every page. Also hyperfilosofical nonsense. It's good but not for everybody.
Haibane-Renmei. Can't recommend this enough. But beware there's no action.
Paranoia Agent just for the surreality. Any Satoshi Kons work actually end up pretty surreal. In a good way.

Death Note! How could I forget to mention that awesome series! I second this recommendation!