Start with picking a home base for your flights, and then work a trip around that. Obviously round-trip is cheaper for flights/cars, but that said, here’s a trip I did if you want to check off a bunch of the “US Must Sees:”
Fly into Reno and get out of Reno as fast as possible. Head over the mountain pass to Lake Tahoe and eat sushi at The Naked Fish, hike, mountain bike, swim in the lake, drive around the lake, etc. Do the long drive South to Yosemite, which is excellent that time of year, and look for abandoned mining towns along the way.
Head South further to Las Vegas (tip: besides hostels which can get you the best advice, company, and prices, consider roadside Casino Hotels: they charge nothing b/c they assume you’re there to gamble for the night).
From Vegas, head over the Hoover Dam and loop up through St. George to camp in Zion National Park. Earn bragging points by hiking Angels Landing. Head SE through Page, AZ (and see the beautiful Horseshoe Bend) on your way to Grand Canyon. At the canyon, get away from the crowds and hike down into the canyon, camping at the bottom if you’re up for it.
Drive north through Monument Valley and camp alone Rte 279 in Moab. Rent mountain bikes and a shuttle up into the mountains, hike in Arches and do slot canyoneering in Canyonlands or the San Rafael Reef.
Drive NE through Fruita (more excellent biking, and an amazing hidden camping spot I can recommend) and Grand Junction, through Breckenridge, past Golden, sip beers with college students in Boulder who will continually say “New-Hamp-Sure-For-Damn-Sure” if you’re from New Hampshire like us, and then catch a flight back from Denver!