I'm a HUGE anime buff.
Stuff I would recommend to any child - Nothing objectionable, etc:
*Akage no Anne - A wonderful of Anne of Green Gables made by Isao Takahata prior to Grave of Fireflies.
*El Cazador de La Bruja: A wonderful epic by Koichi Mashimo.
*Fushigi no Umi Nadia: A work from Hideaki Anno, later notorious for its very complex Evangelion, also somewhat hard, but quite entertaining. Rather weirdly, there's an explicit rebuttal of evolution in a scene!!!
*Blue Seed: A deeply moralistic manga adaptation.
*Seirei no Moribito: A work made in Miyazaki's style.
*Tengen Toppa Gurren Lagann: Somewhat overrated, but still quite inspiring.
Stuff I highly recommend to teens and up:
*Neon Genesis Evangelion
*Gunbuster: Other Hideaki Anno's perfect work.
*Mobile Suit Gundam/Mobile Suit Zeta Gundam: A must-watch.
*Noir: THE Girls-With-Guns anime. Its director, Koichi Mashimo, is deeply catholic and his work often reflects the above. He's an AVID cinephile too...
*Madlax: A weirder creation of the man above...
*Requiem From The Darkness: A very gory horror tale. Still quite pleasant, though...
*Shoujo Kakumei Utena: A deeply bizarre work.
*The Big O: One of the best European-themed animes.
I guess it's a nice introductory list...