I would eliminate Page (hot), Denver and Mount Rushmore (both out of the way) and concentrate on Yellowstone and Grand Teton Nat'l Parks. Both are so different from each other with lots of different things to do in each park. They are both huge areas to explore and you can easily fill a two-week vacation.
In Yellowstone you'll have a lot of slow driving and walking to do. Grand Teton has excellent gorgeous hikes for all abilities to beautiful little lakes. You can take a gentle float down the Snake River in the park and maybe you'll spot a moose in the water with you. We did! For your son (& you, too) there's a chair lift to the ski area, the Jackson town rodeo.
Colter Bay RV in the Tetons has full hookups. Right next door is Colter Bay campground without hookups but has a water fill, dump station and showers. It has 300 sites and none are reservable. Closer to Jackson is Gros Ventre campground in the Teton park and it, too, has 300 non-reservable campsites with water fill and a dump station. You'll see lots of animals in this campground. You will have no problem getting into either of the dry campgrounds without having reservations.
In Yellowstone, Fishing Bridge RV is the only one in the park that has hookups. There are also quite a few reservable and non-reservable campgrounds in the park and many more outside the park. West Yellowstone and the nearby area has a nice selection, including some state parks.
Explore each of the park's websites for campground information and ideas on what to do in the parks and road conditions.
From Las Vegas you could take I-15 north into Idaho to Idaho Falls. Then pick up a beautiful easy river highway, Highway 26 east and take it into Alpine, Wyoming. Then take 89 north into the town of Jackson (good grocery stores and fuel prices) and continue on 89 north into Grand Teton Nat'l Park. Then continue north on the park road which will take you directly into Yellowstone. If you want to get to West Yellowstone outside the park, just continue driving through the national park to W. Yellowstone.
Yellowstone National ParkGrand Teton Nat'l Park
