Hello and welcome to the RTA forums !

A week wouldn't leave you a lot of time to visit a lot of parks unless you are happy to spend just a few hours there as it will take you 4-5 days to travel across the country but a couple of weeks would give you a lot more scope. In Utah the NP's of Zion and Bryce [great for photography] would be top of the list and taking scenic 12 to Torrey, then to Capitol Reef and Arches NP is an option. As you enter Colorado the Colorado NM is an option at Grand junction and this is where you could go South a while to the Black canyon before heading towards Rocky NP. There are many other options and one would be to head to GC once more at the start and then through Page [Lake Powell/Antelope canyon] to Zion. That's about as far as I can go with you with a few suggestions but have a look at the NPS site to find info on any parks that appeal to you. The campgrounds in the NP's are in great surroundings and if you can get available sites I would recommend camping in them. I have no first hand knowledge of KOA sites but to me they seem to be much the same and a bit to "corporate" for my liking.