Hacker Cinema
Cagliostro
1929 · Film · 59 min
Directed by Richard Oswald
Paris, France, 1784. After living many tribulations, Joseph Balsamo, known as Count Cagliostro, an infamous adventurer, enigmatic magician and necromancer, experienced physician and ruthless swordsman, triumphs among the members of the decadent French aristocracy. But a bold foretelling about a very prominent noblewoman causes his fall in disgrace… (Partially lost film.)
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1 comment on Hacker News
I can't recommend a specific order but...
If looking for "adventure" go with Laputa, Cagliostro, maybe Porco Rosso.
If looking for something "warm" go with Totoro, Kiki, Ponyo.
If looking for something more "fantastical" (probably also more complex) try Howl's, Spirited Away, Nausicaä, Mononoke.
If you are willing to consider a TV show, then Sherlock Hound is really fun and delightful in every possible way. The complete series is IIRC 26 episodes.
Also, there are some non-Miyazaki but still Ghibli films which you may want to consider too. Personally, I absolutely love Mimi wo Sumaseba and I can hear the sea, but these may well be a bit uninteresting for a 10 year old; better for a teen probably over 15 I'd guess. Pompoko is clearly recommended for your age bracket though, and it's fun too.
Finally, there's the question of Grave of the Fireflies. I think 15 is probably fine, 10... maybe. But in any case, be warned that it's a very sad story and I have very rarely met someone who could watch it without crying profusely.