First off, HAYAO MIYAZAKI. If you haven't heard of him, under which rock do you live?! The white pebbled one over there or the rocky one I found today in my driveway? THIS MAN CAN DO NO WRONG IN MY BOOK. No joke. He has awesome instrumentals and graphics too. :D
My absolute favorite is Howl's Moving Castle, which is definitely a must-see. The art is stunning, the music is unbelievably catchy in addition to beautiful, and it has solid lessons mixed in with a bit of romance. It may be that I love this one the most because I'm a complete sucker for happy endings and innocent romance - okay, so I rewinded and watched the one quick kiss about 20 times - but that makes it perfect for fans of Disney movies. I included the movie poster below because it has pretty movie snapshots but the DVD cover, which I own, is a bit more bland. This is a movie I'd show my kids (if I had some), though I'd watch out for some bombs. It does say war is very, very, very bad.
If you're looking for more, I'd watch the classics: Spirited Away and My Neighbor Totoro, which I've actually never watched but have heard is great. I've watched many more though and the oldies-but-goodies I'd recommend are: Nausicaa and the Valley of the Wind (snapshot below) and Princess Mononoke. The older ones are a bit more graphic though and shouldn't be watched by a younger audience. Still, if you're up to a little gore, I was surprised by how good the graphics still were.
Another pretty recent one I've watched is The Girl Who Leapt Through Time. It's also a pretty well-known one and one that I thought was pretty good. It can get a bit sad and bit slow at the beginning, but overall the graphics are appealing and there's some interesting originality to the plot.
Here's a quick list of some anime movies that I've heard are EPIC but have yet to watch (if you're interested): Akira, Vampire Hunter D, and Ghost in the Shell. They all lean toward the slightlier heavier side with darker art, so I haven't really worked up the nerve to watch those yet.
\And, finally, what I'm watching now: 5 Centimeters Per Second. It sounds like a heart-rending love story and I'm getting the tissues ready as I type. I've heard it's realistic but oh-so-good. Fingers crossed and I'll report back! I'm also looking into another of Makato Shinkai's movies, The Place Promised in Our Early Days, though I haven't started it yet. It's on Netflix on Instant Play though! ;)