My 9-year old daughter and I share a love of "Sailor Moon". We have a lot of the songs on iPod and she can sing along perfectly in Japanese, remarkable, since we don't speak Japanese at all. We also like Mushi-Shi, and Mermaid Melody.
The only anime series I have watched in its entirety is Neon Genesis Evangelion, which blew my mind.
I am big into anime films though - namely Miyazaki (of course) and Satoshi Kon (Paprika, Tokyo Godfathers, Millennium Actress). Grave of the Fireflies was also excellent, but absolutely devastating and difficult to watch more than once.