Southern Utah is another wonderful place to visit. Here are a few links to some of the sights to see there:
Map of Utah
Kodachrome Basin State Park, Utah
Grand Staircase-Escalante National Monument, Utah
National Park Guide
Monument Valley, Utah/Arizona
Arches National Park, near Moab, Utah
Utah National Parks Introduction, Southwest USA
A couple of good places to set up a base are Kanab and Moab, both of which have several nearby parks and monuments. Kanab is close to Bryce Canyon, Zion, Escalante-Grand Staircase, Cedar Breaks, and the North Rim of the Grand Canyon. Moab is close to Arches, Canyonlands, Monument Valley, Dead Horse Point, and Four Corners. Between these two towns are Capitol Reef and Lake Powell, as well as several other parks, forests, and monuments.
This is one of my favorite areas to visit, but is certainly not the only place to consider. Others have already mentioned several others, though, so I thought I'd add this area for your consideration. Las Vegas, Nevada is only a few hours from Kanab, Utah, so it might be a good place to start a trip through this area. If you travel from Las Vegas, you could also see the Virgin River Canyon in Arizona on your way to Utah. It is spectacular, in my opinion.