I would suggest watching アニメ 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!!
あなた によって no means have to watch all of these. They're just rather 人気 アニメ that most everyone in the アニメ world knows at least something about.