I'm generally a really big fan of anime. As for what kinds, I generally prefer Shonen anime, and generally ones with science fiction or fantasy aspects. I also prefer anime with no fanservice, or at least minimal fanservice.
I currently watch: Naruto, Bleach, Moribito: Guardian of the Spirit, Scrapped Princess.
My favorites of all time are: Fullmetal Alchemist, ScRYed, Noein: To Your Other Self, Samurai 7, as well as Naruto, Bleach and One Piece.
Other ones I've seen and liked (I'm missing several, I can't think of them all) are: Blood +, Yu Yu Hakusho, Arjuna, Argento Soma, Shingu, Cowboy Bebop, Outlaw Star, Trigun, Blue Gender, Kenshin, Shaman King, Getbackers, Digimon: Digital Monsters, Death Note... and others.
I also loved Teen Titans and Avatar: The Last Airbender, even though they were "American anime."