'ROBMAT' in a previous post gave you some great ideas up to Yellowstone. Each national park, national monument and national historic place has as wonderful web site which will give you lots of good information such as things to do, road maintenance issues, camping information and more. Check them out. For example, here is Yellowstone's:
Yellowstone National Park (U.S. National Park Service)
I'd highly recommend you stay in the parks themselves for the best overall experience. They have ranger programs for the children and nightly campfire programs. They also give guided walks/hikes.
From Yellowstone your could head north to Glacier Nat'l Park or head south to Grand Teton National Park which is attached to Yellowstone but is completely different in looks and things to do. It's like a miniature Alps! The hiking is better in the Tetons than Yellowstone as they lead you to lovely lakes and streams and they are for all abilities and lengths. Definitely plan a gentle raft trip down the Snake River in the park. You just might see a moose in the water with you....we did.
Then if you're ready to return home continue south into Utah. All the national parks in this state are awesome. Although the summer will be hot in many of the parks, plan to do your hikes and siteseeing in the morning and again around 3pm. Noon to 3 is siesta time in the heat. Bryce Nat'l Park will be the coolest. If you make it to Arches Nat'l Park definitely sign up in advance for the Fiery Furnace ranger-led hike. It is so unique and the whole family will love it!
If you still have time you could then visit the Grand Canyon. The North Rim is much cooler but for first time visitors I recommend the South Rim. Hike down the Bright Angel Trail if only for 1/8 mile or so and you'll get a completely different feel for the canyon rather than staying on the top. Trailer Village RV in the park is the place to stay if you want hookups; Mather campground if you don't need hookups. The free shuttle stops at both campgrounds and you can use it all day long getting on and off where you want. This is nice for doing the Rim Trail which is reasonably flat. Walk a while then pick up the shuttle again.
From the Grand Canyon, Flagstaff has some nice Indian dwellings to explore. We like to stay at Bonito national forest campground which is right at the entrance to Sunset Crater Nat'l Monument - no reservations; no hookups but if you get there in the morning after leaving Grand Canyon you'll get a site. It's lovely.
You'll also be close to Sedona, AZ which is beautiful, especially if you can walk a trail for a mile or so and get away from the crowds. At that point you can head home via I-40 or drop down to I-10/I-20 for your return home.
One summer trip just won't do it for you. You'll be back! Have fun planning and let the children help, too. Perhaps tack some state maps up and let them research different things they'd like to see.