hmmmm didnt know so many anime fans were on this board... i firmly believe its a format of media and not just a "genre" so it really depends on what you are looking for....
anyway, a lot of good ones mentioned but i could recommend a few more:
slow pace, deals with oddities such as spirits and paranormal (not scary stuff), almost mockumentary-like:
-Mushishi
-Natsume Yuujinchou
slice of life, comedy, sitcom-like **girly stuff**:
-Bakuman
-K-On!
Scifi:
-Haruhi Suzumiya
-Guilty Crown
Funny Slap Stick:
-Space Dandy
-Gintama!
Shounen (DBZ/Naruto-like):
-JoJo's Bizarre Adventure
-Soul Eater
Sports Related:
-Kuroko no Basket (basketball)
-Haikyuu! (volleyball)
Mind-fuc*ed/thinking:
-Hikaru no Go
-Death Note
Movies that are good:
-Grave of the Fireflies
-Summer Wars
-Girl Who Leap Through Time
-Paprika
-Princess Mononoke
WTF satirical/fables/play on words/etc:
-Monogatari
-Katanagatari
Shoujo Magic Girls:
-Madoka Magica
-Magical Girl Nanoha
For toddlers:
-Folktales from Japan
-Doraemon
**Movies for kids to watch - Hayao Miyazaki eg Ponyo and Spirited Away**
All these of i have mentioned are worth your time. There are lot to watch but i feel like these are pretty good to start off with their respective genres if you like the those genres. My favorites, of course, are shounen jump and slow placed like so I really enjoyed Full Metal Alchemist Brotherhood and Mushishi.