I'd agree with Akira and, from Studio Ghibli, I'd take Grave of the Fireflies (an amazing film; there's something about the separation that comes with animation, even when dealing with real subjects and situations, that both detaches us and makes us put ourselves in the protagonists' shoes) and either Mononoke Hime (Princess Mononoke) or Spirited Away (I like Mononoke because it's nice to look at the film from the perspective of 'there are no bad guys, just people trying to live out their existances either with or against nature... people trying to survive'... plus Neil Gaiman wrote the adaptation). My favorite Ghibli films are Porco Rosso and The Castle of Cagliostro, but there isn't much meat to either of them except for a ripping good yarn (the former's animation is beautiful and the latter is one of the best English dubs I've ever heard). It depends on how you do want to approach animation since there really is a huge selection.
Iron Giant was a film that didn't receive enough kudos when it came out. It was a very heartwarming story with pretty good voice acting and animation. I don't cry much at films, but I teared up during the "Superman" line.
If you're going for weird, check out Bill Plympton's I Married a Strange Person.
My husband and I also really liked Tokyo Godfathers. I liked how the plot threads interacted with each other in the story; it's full of sadness, beauty, and hope.
Looking outside both Asia and the US, I'd recommend a gander at The Triplets of Belleville, which is the type of movie that you can find yourself dancing out of the theater to it's joyfulness.
It would be interesting to hear your thoughts on the old Fantasia with the new one.