Unless you really want to drive through Denver (traffic) for some reason, I'd recommend you come in via I-80 then head south into Colorado to Fort Collins and take Hwy 14 west. Pick up Hwy 40 around Steamboat Springs and head west to Dinosaur Nat'l Monument. It's a beautiful drive and will gradually introduce you to some mountain driving.
https://www.nps.gov/dino/index.htmThen continue north from Vernal, UT on 191. You'll immediately have a steep uphill toward Flaming Gorge Nat'l Monument but your rental isn't big and you shouldn't have any issues. You'll just be going slow uphill. When you have to go downhill use your gears, not your brakes. Shift into a lower one at the top before you get going fast midway.
Take 191 into Jackson, WY (stock up on groceries and fuel) and on to Grand Teton Nat'l Park where you should plan on stopping for a few days. The end of August you won't even need reservations especially if you can dry camp... and you can do so easily. Fill your fresh water tank and empty your grey and black tanks just before entering the campgrounds. During your stay conserve water. There will be shower facilities to help you do so. The first campground you'll come to is Gros Ventre (our favorite). Ask if one of the few electric sites are available. Otherwise you'll have a dry site. Continuing north through the park you'll then come to Colter Bay campground. Each of these two have 300 sites and rarely fill. Right next door to Colter Bay campground is Colter Bay RV park and you just might get lucky to get a site there if you want hookups. Grand Teton is a lovely park. The trails are awesome and even the easier ones will take you to little mountain lakes. The boys would love the gentle raft trip in the park down the Snake River. You might even see a moose in the water with you. We did!
https://www.nps.gov/grte/index.htmThen continue north taking the South Entrance into Yellowstone to your camp destination.
You'll have a wonderful trip!