If you pass through the Salt Lake City area...
Travel your way down to Huntington or Castle Dale Utah. You are going to want to go to the Wedge Overlook (Google Maps: Rd 332 from Huntington, or Rd 401 from Castle Dale). Both roads are the same condition, with 401 being a tiny bit wider but both allowing 2 vehicles to pass each other. These are maintained dirt and you can get going a good clip across there.
After the Wedge Overlook, continue East to the Buckhorn Draw road (rd 332), and go south. This will take you past a few petroglyph walls, and continue on through a canyon till you cross the Green River (there is a bridge). Then continue on to the I-70 interstate where you can head east toward Moab. There is camping at the Wedge Overlook and in many pullouts along the Buckhorn Draw Rd. Many spots, just find one and pull in. We pitched a tent inside the cave on Buckhorn Draw last fall.
You can also continue on via dirt road (Temple Mtn Rd after you cross I-70) to the Goblin Valley State Park (also accessible via paved road from I-70). From here, you can choose to go south to Hanksville/Blanding then back north to Moab (longer route, but way way more scenic!!!), or go back to I-70 and go south in to Moab (shorter route). If you've got the time, go the long way!
Are you going to be passing through Missoula Montana? I did this route in my truck last fall (slept under the tonneau cover on my way to Canada).
Go south out of Missoula to Grantsdale, take the 38 East; the 38 is maintained dirt with a paved section at each end. Good views, some minor switchbacks, passes you by a waterfall right next to the road. You could get a 20-something foot trailer over that road no problem. Continue on to Anaconda then go south on the 569 (paved) on the East side of Anaconda and go to Wise River. At Wise River, go south on the Pioneer Mountains Scenic Byway (paved), and will take to past the Coolidge Mine (accessible via good dirt road, plenty of boondock camp opportunities, but the mine is a little bit of a walk from the parking area. You can ride dirtbike, 4 wheeler, SxS, or bicycle to the site though, just no Jeeps/trucks). Then continue on south and get on the I-15. This route keeps you in the mountains in the cooler temperatures for the summer months. I drove this September last year, and camped just south of the Coolidge Mine, was in the upper 40's in the morning if I remember correctly, and didn't sweat at all during the walk around the mine.