"If any place south of the Alaskan tundra qualifies as America's Serengeti Plain, [Wyoming's Red Desert] is it."
— Frank Clifford, Los Angeles Times, May 5, 2000.
The Red Desert's six millions acres (slightly bigger than Vermont) is home to over 350 different species of animals, including the largest migratory game herd in the lower 48 states (pronghorn antelope) and the largest desert elk herd in the world. Nothing to do with animals, but interesting nonetheless, the Red Desert also has the largest active inland dune system in North America.
Six million acres is a lot of land to cover in your quest to see some wildlife though. Take a bite-size chunk by concentrating your efforts around Oregon Buttes and Steamboat Mountain, outside Farson off Route 28. (Make sure to pick up a Wyoming Gazetteer so you know exactly where to go.)
Dawn and dusk give you the best chances of spotting pronghorn antelope, mule deer and elk. The area around Steamboat Mountain (which happens to be an ancient Native American buffalo jump site) is smack in the middle of the migration areas for all the big game species. If you're lucky, you might even spot a herd of wild horses.