Las Vegas to the 5 National Parks in Utah to Yellowstone

We just completed a visit to Zion, Grand Teton and Yellowstone - something we have experience doing. If I had some recommendations, they would be this: We always use Jordanelle state park as our "go-to" place. It is about 10 miles outside Park City, UT (also a great place to visit while you're in the area). Driving from Phoenix, we use Jordanelle as our stop-over on the way to points further north, then also on our way home. It is among the nicest facilities we've found in this part of the country. We stayed at Red Rock RV Park in Island Park, ID for our visit to Yellowstone this time. It was about $60 a night, but worked well for us due to vast open areas, and very rural, quiet atmosphere. We stayed several nights in Gros Ventre campground inside Grand Teton NP. It was bearable because it was largely vacant and we had no one else camping around us. The place is run down and largely in a state of disrepair. Nonetheless, we had fun and moose are commonly strolling the campground. We enjoyed some whitewater rafting on the Snake river in Jackson while there (Dave Hansen Whitewater Rafting Company). Awesome experience. We stayed a couple nights at Bryce Zion RV park in Glendale, UT. Nothing to write home about, but it worked for us. Horses surround the campground and our girls enjoyed that aspect. It was quiet and more spacious than Bauers RV park; which came highly recommended but was in town and seemed too tight-quarters for a family with dogs. Have fun with your trip. We sure did!

Here is a recent post that should help with the UT part plus in my second answer some CGs and a route to the Tetons. I've made this trip in a 42' MH. There is a tunnel between Zion and Bryce that's a problem for big rigs. Most will not recommend taking 12 between Bryce and Capt Reef although I have. There are some short very steep areas and it's kind of curvy. These are the only areas of concern all the way to Yellowstone.
If you have the time I'd recommend the N Rim after doing Zion from the west side. 59/389/89A to Jacobs Lake will be 60 miles down and back but then you do not have to deal with the Zion tunnel. There is a USFS CG at Jacobs Lake and a private CG so do the rim in the PU from there. Leave the N Rim just after sunset and see how many mule deer you can count on the way back.