Basically any movie by Hayao Miyazaki will be good. My recommendation goes in this order:
My Neighbor Totoro
Spirited Away
Princess Mononoke
Laputa: Castle in the Sky
Howl's Moving Castle
Nausicaä of the Valley of Wind
Kiki's Delivery Service
Whisper of the Heart
Porco Rosso
All these are good movies, and cand stand up to comparison with any big-name movie. Some of them are even on IMDB's top 250 list, which I consider a great achievement. And best of all, they work for almost anyone, boy or girl, young or old, almost regardless of taste. They are the perfect entrence to anime.
Other than that, I'm bad with shoujo anime, since I'm a boy myself, and I've noticed that girls more often like that type of anime. I guess it depends on the taste of your friend. I would not recommend Naruto to a 25 year old that likes romance, but I would recommend it to a 14 year old with a love for action and comedy.
Manga you have to read (unsolicited recommendations):
Yotsubato - funniest thing I ever read
Chii's Sweet Home - the cutest cat-story you could imagine, and my avatar.
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