I would suggest watching anime 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!!
You by no means have to watch all of these. They're just rather popular anime that most everyone in the anime world knows at least something about.