Our travels to all these areas have been in our RV. We first visited Yellowstone and Grand Teton in September 2010. We spent two weeks parked outside of the park, in West Yellowstone. Then we moved to Grand Teton, which has a campground in which we can fit, and finished out the season there. Note that there is a fourteen night limit inside the parks. Now, this was during bad economic times, so we had no trouble with reservations. We learned about altitude sickness, which slows us both down.
Autumn was beautiful, the snow at higher elevations was rather intimidating, as this was our first time in this ecosystem. It's mating season for the elk, the males are magnificent. We did smell wildfires, although they were supposedly of the controlled burn type. They still stink, they still compromise views, no matter the source.
The next time we went to GT in July 2011. Winter had been severe, many a trail was still closed to hikers unless you had winter gear. The wildflowers, well, it looked like Monet himself had been there painting. It was crazy with tourists, everywhere. We stayed the entire fourteen nights allowed. We had made our reservations as soon as was possible, which I believe was right after the campgrounds all closed up in late September.
Our third visit was in June 2015. Schools were out for summer break, but mostly it was still childless visitors. Again, the wildflowers were stunning. Again, we stayed the maximum of fourteen nights. And again those reservations were made the previous autumn. We got a hotel one night up in Yellowstone in the Mammoth Hot Springs area, and spent some time driving places we hadn't accessed before. Smiling remembering the babies, all the babies. Bison calves galore in the Lamar Valley area. Elk calves right up in the hotel area. Yeah, smiling remembering.
We did the Utah Five in late April through May in 2013. We spent two weeks outside of Zion, then moved on. It was not crowded in most places, yet. The exception was Moab, from which one can access both Arches and Canyonlands. The city of Moab has a very active events schedule. We had a very hard time finding a place to stay there. Utah 12 is indeed an amazing drive, especially when viewed through the windshield of a motorhome. Just don't look down in places, gulp. Utah 24 is an otherwordly drive. Spring was already getting warm at the lower elevations, not so much at the higher ones.
From Moab it is an easy drive to Colorado National Monument in western Colorado, which is what we did while waiting for a spot in Moab. Afterward we went to Mesa Verde, which is also quite lovely.
Please think about taking your time as opposed to rushing through all these treasures. Remember the philosophy here, assume you can return.
Many a time in our explorations of the west we've gotten chased around by wildfire season, and it's getting worse and worse. For this reason I would recommend spring, as places are still in their green stage and are less likely to be burning.
You said you enjoy camping. A potential option is to rent an RV. We saw many a unit from Cruise America (and it's counterpart, Cruise Canada), as well as El Monte.
I know the park service is undergoing changes, but it looks like the annual pass is still available, as is the senior pass. We never minded the eighty dollars each year, considering it money well spent.