Seconding/thirding Fruits Basket, I never watched the original or read the manga, but I'm following the new anime and it has plenty of heartwarming moments. My ultimate heartwarming anime, though, is Hidamari Sketch - particularly later seasons just make me melt from all the warm and fuzzy feelings.
Another anime I found very heartwarming that I don't think anyone listed yet is Princess Jellyfish (I gotta rewatch it again, it's been several years...)