Slice of life, fantasy, mystery.
Can you suggest one each?
Slice of life - Silver Spoon, Space Brothers (Can be streamed on crunchyroll)
Fantasy - One Piece, Naruto, Bleach; pick either one that looks most interesting to you. I personally find Naruto to be the best. (Can be streamed on crunchyroll).
Mystery - Death Note, Monster (haven't watched yet, but is supposed to be really good), and of course the legendary Master Keaton. I dunno where these 3 are streamed, but Master Keaton or Death Note would be good first time things. Master Keaton is basically Sherlock Holmes based in Japan,
Edit: For fantasy, add Moribito to my list. It's more historical, but it's a really well done, classy show. You can watch it on crunchyroll too.
Edit 2: Slice of life - if you like music, try Kids on the Slope. If you also like action/fantasy, try Samurai Champloo