Ken, if you haven't already, I'd suggest you start off by visiting Mt. Rushmore in the Black Hills of South Dakota, following that up with a visit to Devil's Tower in NE Wyoming nearby. Continue to Yellowstone NP, then down to adjoining Grand Teton NP. You will be close enough to do the Grand Circle of NPs in Southern Utah, but be advised that it will soon be exceedingly hot there (no kidding), so check the extended forecast before deciding whether to do that or not. You could always head North instead, to Glacier NP, and even North Cascades NP in Washington. If time is still available, you could add in Olympic NP in NW Washington.
That sounds like a pretty ambitious schedule there for the allotted time. These are just favorites of mine since I live in this part of the country. Others have their own personal favorites, so keep an open mind. Above all, take your time, be safe, and enjoy the trip. Oh, even at this time, finding available campsites in the NPs is going to be difficult, so plan on alternatives!