My family of 5 crossed the Mississippi for the first time three summers ago from N. Carolina. (28' TT bunkhouse) Best trip ever, but we did have the good fortune of having 3.5 weeks. As mentioned above, SW South Dakota is a great place to start. The Bad Lands are awesome and Mt. Rushmore and everything nearby (including Custard Park) are great. We then went to Cody to see a rodeo on the way to Yellowstone, which is a great trip along with the Tetons, but time may be an issue as each would take a couple days minimum.
Were are planning our next western adventure for next summer, Grand Canyon, can't wait!
A couple things we learned. Take a couple of spare trailer tires, you can always return if not used, but we went through 5 between two campers (in-laws traveled with us on their camper). (btw plastic levelers work great as jacks for changing tires, just stack up and roll the good tire up assuming you have dual axels) Also, get off the interstate some, great roads and awesome scenery in the west.