The best general advice I've discovered is in Lisa Gollin Evans's book An Outdoor Family Guide to Yellowstone and Grand Teton National Parks (Mountaineers Books, 2006). Older children can learn about nature by enrolling in the Yellowstone Junior Ranger Program. The Young Naturalist program at Grand Teton provides children with a similar opportunity to explore the natural world of the park and costs $1.
Especially for Kids
Yellowstone has a Junior Ranger Program for kids ages 5 to 12. The $3 fee covers the cost of the activity paper, Yellowstone's Nature, which describes the requirements for attaining Junior Ranger status. Activities include attending a ranger-led program, hiking, and keeping a journal. When participants complete the program, it's announced to the public with great fanfare, and a patch is awarded. Sign up at any visitor center.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.