Your going to need a vacation when you get home...
Firstly, I live in the western, intermountain region of the U.S. The trip you have considered from Vegas to Salt Lake (as the natives call the City) an onwards will cover immense distances with very little between the desired sights. Vegas to St. George is a couple of hours, most of which is in the bottom of a canyon on Interstate 15. While, initially it's quite beautiful, after a while it's quite boring. After emerging at St. George, it's high desert all the way to Provo, some 60 miles or so south of Salt Lake. If you were going to consider Yellowstone, the best route then, would be to go north from Salt Lake into Idaho and just north of Idaho Falls, turning east bound to come out at Jackson Hole, Wyoming. It's on the south boundary of Yellowstone. if it hasn't snowed by that time, you can exit Yellowstone east bound and come out at Cody, Wyoming, where you'll need to spend at least a day going throught the Buffalo Bill Historical Museum and Whitney Gallery of Western Art (has the world's largest collection of original western art, including Remington and Charles Russell.)
Then its' about a 10 hour drive across the Big Horn Basin of Wyoming, across the Big Horn Mountains, across the Powder River Basin and through Sundance, Wyoming to Rapid City, Sount Dakota for the Mount Rushmore sculpture and Black Hills National Forest. I'd consider getting on the airlines in Rapid City to Denver, Colorado to make my way back to Vegas (you can rent cars one way for an additional fee. Otherwise you'll still be travelling this time 2010... Best of luck and welcome to the U.S.!