Have done this trip a few times in both one day, non stop travel as well as the scenic route hitting the parks along the way. The quick way is 14 hours or so once you cross at Coutts. Travelling the I25 through Wyoming is 400 miles of nothing, will make for a quick trip home. You could consider a detour over to Devils Tower its about two hours (each way) off the interstate but pretty neat.
https://www.nps.gov/deto/index.htm
On your way down, try to stop by the Lewis and Clark Caverns they are pretty cool and both you and the kids should enjoy. Give yourself a couple of hours here, and take the tour. Its nice stop to get out of the car and see something that will likely leave the kids in awe.
https://fwp.mt.gov/stateparks/lewis-and-clark-caverns
Yellowstone is a must, these accommodations, right at Old Faithful are both rustic and affordable, even a little fun for a night or two.
https://www.usparklodging.com/bookin...hp?prop=100200
In July Yellowstone can get pretty hot during the day, but cools down nice in the evenings, really busy but to be expected. One day for the north loop and one day for the south loop should give you enough time to see pretty much all you could want to in the park.
From Yellowstone you can head right south to Grand Teton, not a lot to see except for the Tetons themselves IMO. Lodging options below.
https://www.gtlc.com/lodges/colter-bay-village
Jackson; I describe it like Banff meets Taber, worth checking out for an afternoon. I like this Mexican restaurant there which, fun fact, has a small role in the Brian Laundrie, Gabby Petito case.
https://www.merrypiglets.com/
Leaving Jackson and making your way to Denver is a long drive and have not found much of anything to see or do in this stretch, you will likely just wat to get to Denver at this point anyways. Outside of the NW, Wyoming is a dump, as is are areas of Montana you'd be driving through. They longer give tours of the Gibson guitar factory in Bozeman MT, regret that we never stopped to do this.
The Utah parks mentioned above are really a whole other road trip, I recommend seeing them all but are quite the detour and time commitment to squeeze into a trip to Denver.