Jan,
A route which provides spectacular scenery, little traffic, few tourists, and good weather in September is highway US-89 from Yellowstone National Park to Flagstaff, Arizona. The route passes through or near Yellowstone, Grand Teaton, Bryce Canyon, Zion, and Grand Canyon National Parks and many other scenic areas.
The only large city on the route is Salt Lake City, Utah, otherwise most of the route is through fairly remote areas and small towns. It would give you a real feel for the western U.S.
The north end of Route 89 is at the Canadian border near Waterton-Glacier Park and Glacier National Park, however it's a long way from there to Yellowstone. You could fly into Bozeman, Montana, to begin at Yellowstone.
September is the ideal time to ride in most of the U.S., including Route 89. Nights would likely be chilly in Yellowstone and Teaton parks, but the weather is normally mild further south. Traffic and park visitation is light in September, because schools are in session.
I've cylced the route and enjoyed it greatly. The April 2007 issue of National Geographic Adventure magazine has an article on driving Route 89, which describes the road and places along it. The article is available on the magazine's website.
If you'd want to ride futher, you could continue to Phoenix or Tucson, Arizona. The Mexican border is about 100 km south of Tucson.