Well it can be good to get some variety. Since you have seen Cowboy bebop I would asume that you have mostly seen dub. Death Note is a good thinking anime, can be a bit dark and it may be a bit confusing with whether you suport the main character, but it is good. Hellsing is pretty good, it is best to watch the TV series then watch the OAVs, it is a bit gorey but is a good watch. If you want to go for something a little more romantic, Chobits is a good watch, might look like a romantic anime that covers a relationship between a human and robot might not go very deep, but this also covers the topic is it right, and that people are losing relationships between each other, also good look at Japanese society. If you enjoyed studio Ghibli a movie you might enjoy is The Girl Who Leapt Through Time which can have a simular feel, though I enjoyed it a little more then them.
One of my favirout anime companies is Kyoto Animation, which has many good anime, my favirout being The Melancholy of Haruhi Suzumiya, which is a very popular anime (don't be too freaked out over certain things and it is best to watch in original broadcast order not the chronological order). Another one is Full Metal Panic though first season isn't done by KyoAni, it is a goood mix between mecha and school life, though it is mixed up in second season (Fumoffu) where it takes the battle field into the school yard (paroding itself, and the back for The Second Raid., by the way English dub makes me cringe. There are some good slice of Life; likeLucky Star there is no plot, but can be an easy watch with nice humour with anime references, and the moe fest K-ON! that recently just finished, also an easy watch. And though it might feel not feel like a good genre, the romantic genre can be quite good, it is a nice idea to start with Kanon 06 version, then watch Clannad, this is because though both are very funny, both can be very sad and some experience in Kanon may be good before the depressing moments that especialy appear in Clanads sequel Clannad: After Story, they are both masterpeices.
Some over good ones are Zero no Tsukaima, Shakugan no Shana, Toradora! and Hayate no Gotoku each has a simular tsundre character and if you like one you might like the others, but each has its own different merits and especialy Hayate is funny. And if you want maybe Sayonara Zetsubou Sensei, I could liken it maybe to the anime version of Family Guy, but is actualy witty, the characters each have some mental problem, and is very good at looking at problems in society and making therological questions.
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Through dream I travel, at lantern's call
To consume the flames of a kingdom's fall