I lived in Japan for 7 years throughout my 20's, during which I was converted over as an anime/manga fan.
Since so much has already been covered, I'll address a few I haven't yet seen talked about.
I'd have to say that Boku Dake ga Inai Machi (lit. "The Town Without Me", English title "Erased") is probably the best anime made in the last year. It combines several aspects of different genres, primarily slice of life, time travel, and a murder mystery to boot. I'd compare it to Quantum Leap, except the traveler is leaping back to events and moments in his own life.
If you're in for movies: Colorful and Grave of the Fireflies are on my "must see for every anime fan" list
This character that I'm writing was heavily inspired by the main character from Jormungand, which I'd recommend for a more action oriented series. And another in the sci-fi realm is Shingu: Secret of the Stellar Wars.