Most of my suggestions are already really popular. :)

Death Note, Cardcaptor Sakura, Tsubasa, Vampire Knight... Sailor Moon. It's not really like the anime in a lot of ways (especially the dub), so even if you don't like Sailor Moon, I think you should at least take a look.

My one suggestion that probably most people don't know? Mermaid Melody Pichi Pichi Pitch. All I knew about it at first was that it's a magical girl series about mermaids whose power is singing, but... it's so much better than just that!