Ubud
Take walks in the rice fields - My kids loved walking through the rice fields. The elaborate irrigation ditches take hours of investigation and baby ducks paddling in the water are adorable. Here in rice fields, life is lived as it has been in Bali for centuries. It's calm, peaceful and the kids can stretch their legs. At the bookstores, buy the map "Ubud Surroundings" (by Travel Treasure Maps) that marks all the paths through the fields.
Tourist dance performances and shadow puppet plays - Having lived in Bali previously, I thought I'd take my boys to see "real" Balinese theater performances in the villages late at night. As it turned out, they liked the tourist performances better, especially the Wayang Kulit (shadow play). These tourist performances often have first rate performers, but the show will typically start at 7:30pm and the story is condensed. Check the tourist office in Ubud for times and locations of Wayang Kulit, Legong, Kecak.
Puri Lukisan (art museum) Barong dance - On Sunday mornings, there's a very funny and enjoyable Barong dance performance by kids. Sit out on the grass and watch this performance where the kids play the gamelan and act all the parts.
Take Balinese dance and music lessons - Kids in Bali are accomplished dancers and musicians and your kids can get into the swing of things with dance and music lessons. Children have flexible limbs to stretch their hands into those lovely gestures, or take to the rhythms of Balinese drumming and "xylophones." At the Pondok Pekak Library and Resource Centre (east side of the soccer field, off Monkey Forest Rd.) they have dance classes for kids and a kid's gamelan (plus a children's library with books in English).
Goa Gajah (Elephant Cave) - Goa Gajah doesn't have any elephants, but it does have a wonderful witch with droopy ears sculpted in stone over the cave entrance. Going into the cave is like going through the mouth of the monster. The cave complex was a sanctuary for hermits and a sacred watering place.
Ayung River - If your kids are 8 and over, take a water rafting trip down the Ayung River. It's a great way to see this deep, lush river gorge.
Alternatively, you can walk on paths along the Ayung River in Sayan. The paths are somewhat steep, but marvelously jungly, bridge crosses over to Bongkasa village.
Monkey Forest - It's fun to see mischievous monkeys scrambling and scurrying around the immense roots of the banyan trees and lush tropical vegetation. You can certainly see the monkeys up close - they may come up to pluck at your legs, but please don't feed them.
Herons at sunset in Petulu - Visit the white heron sanctuary in Petulu, just north of Ubud. Go at sunset to see a spectacle, thousands of birds coming home to roost. (Careful of bird droppings.)