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 (The Mountaineers; www.mountaineersbooks.org). Another useful guide to consult is Frommer's National Parks with Kids, by Kurt Repanshek. Both guides are easy to find in bookstores.
Older children will be entertained and receive a nature lesson by enrolling in the Yellowstone Junior Ranger Program (no specific age restrictions) at any visitor center or ranger station. The $3 fee covers the cost of the activity paper for Junior Rangers, 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.