There are four tunnels along highway 16 that point directly at Mt Rushmore.
Yellowstone is another great place to visit, so is Glacier Park. Glacier you can not drive a motorhome over the Going to the sun road, but I took my car over it and took a tour bus another year. It is close to the border crossing into Canada. There are very steep mountains, and well worth looking over it.
Yellowstone, I entered from I90 near Livingston. You can enter several ways, say from Grand Teton NP in the south, going north into Yellowstone, or Livingston, or West Yellowstone. If you really like mountain drives, and don't mind paying a fortune for gas to go over the 10,000' (3,100 meter) elevation pass. The other ways into Yellowstone have much better roads. I even went from Cody to Yellowstone (well went from the center of Yellowstone to see Cody, then came back to Yellowstone) and that was a OK road too. (but that was back in 1994, I recall it has some really steep hills).
You might want to leave LA (after Disneyland, Sea World, and Universal Studios) and then head east towards Williams AZ, where the Grand Canyon is. You might want to go east as you leave the Grand Canyon, and then head towards Monument Valley. That is where the 'mittens' are located. Look at the picture, you will see the mittens.
From Monument Valley, going north toward Moab, you can see lots of rocks and canyons. Along highway 128, toward the Colorado River, near Moab, I got a lot of wonderful pictures early in the morning, as fog was banked up against the rock formations. Arches NP is also there. Google both.
From Moab, you will be directly south of Jackson Hole and Yellowstone. You probably do not want to criss cross the countryside at $0.50 per mile fee to the RV rental company, not to mention gas cost is another $0.50 per mile. Glacier NP is northwest of Yellowstone. Canada is north of Glacier.
How do you plan on returning the RV? Back to LA? IT might be less expensive to return it to another location (if they have them) such as Salt Lake City and then not need to pay the 600 miles in gas and mileage charges to take the RV all the way back, and just fly there.
You might also want to consider Yosemite NP and Sequoia NP. One of the very high waterfalls (1400' or 400+ meters) the other for the large trees (70M) and near the south entrance station of Yosemite, there is a tree that is 10M diameter and about 60M tall. The first branch is 30M high, and about 2M diameter where it attaches to the tree trunk. Look for the "Grizzly Giant" tree. Sequoia NP is higher elevation - about 5500' - 6,500' elevation (about 2000M while the valley floor of Yosemite is only around 2,600' or 700M. Tioga Pass is about 10,000' elevation, while Mt Whitney is 14,550'. Lots of Meters high!
Perhaps the return trip might take you past the Grand Canyon, while the northbound trip can take you past Yosemite, Sequoia NP (they are only about 50 miles apart) and then north to I80, then east to I 15, then north to Yellowstone, Jackson Hole and Glacier NP, then on to Canada?
Lots of planning, good thing you will not be going this month!
Have fun camping!
Fred.
