I would suggest watching 日本动漫 that a lot of people know to test the waters and see what genres you're into.
A few that I would suggest would include...
Death Note Fairy Tail Ouran High School Host Club Fullmetal Alchemist Vampire Knight Black Butler Soul Eater Sword Art Online Attack on Titan Code Geass The Melancholy of Haruhi Suzumiya DuRaRaRa!!
你 由 no means have to watch all of these. They're just rather 流行的 日本动漫 that most everyone in the 日本动漫 world knows at least something about.