I wish I could take a trip like that myself this summer.
Some basic suggestions come to mind--
The eastern half of the country will mostly be miserably hot and humid during the summer. The West will just be hot, and cooler in the high country. My own bias is toward the West and outdoor scenery.
Pick a place to fly in and out of, and make a driving loop in that region. (Northwest, central Rockies, Southwest with California, for example) That will let you see a lot, but put a reasonable limit on the amount of driving.
The western states are BIG. It will take most of a day to cross one, especially if you stop at the sights or walk some trails.
If you plan to stay in any of the national parks, it would be a good idea to make reservations well in advance. At the first park, buy an annual pass--this will cover entry fees at national parks and many other federal sites.
Annual Pass
Here's one suggested itinerary, with a focus on natural scenery and Native American\Western history:
Fly into Denver, and make your way southwest to Mesa Verde National Park.
From there, go northwest to Moab, Utah. You could spend at least two or three nights there to explore Arches and Canyonlands NPs. Canyonlands is an hour's drive from town, and Arches has several shorter trails. (As an alternative, go southwest from Mesa Verde to Monument Valley and Grand Canyon, and then back north.)
From Moab, make your way north to Grand Teton and Yellowstone. Plan to spend at least one night in each, if not more. From Yellowstone, there are plenty of historical and natural features to see throughout northern and central Wyoming as you make your way back to Colorado.
If there's time, go up to Rocky Mountain NP and the town of Estes Park before returning to Denver. Lodging might be reasonable at the YMCA in Estes Park.
Enjoy your trip!