Where to camp in Nevada, Arizona, New Mexico

In Taos I would 2nd Taos Valley RV Park or you could also try Taos Monte Bello RV Park out towards the Gorge bridge but hear it is usually windy out on the mesa/.
Santa Fe we used Trailer Park Ranch, in town but very well kept.

For the Grand Canyon, the Grand Canyon RR RV park Williams AZ is the best. Walking distance to downtown Williams and a nice brewery/restaurant across the street. Taking the train to GC (you are camping right by the station) is really fun and a way to avoid parking headaches. If you travel with kids, there's a Deer Ranch about 10 miles east of Williams. About an hour and a half West of Williams on the old Rt 66 alignment west of Seligman is the kitschy Grand Canyon Caverns. A step back into 1950's tourist trap stuff.... Also have an RV park there.

If your travels take you to Moab, be sure to take highway 128 that runs around the east side of Arches Park to get to/from I-70. One of America's secret spectacular rides.

For Valley of Fire I'd suggest Sam's Town Casino in Henderson (South Vegas). Easy access for big rigs. Nice adjacent Casino with good restaurants and free shuttle to Old Downtown and the Strip.

My wife and I have traveled quite a bit in some of the areas you mention.

Moab - In Moab, we have found the Portal campground to be the best in the area. Clean, quiet and close to Arches. The big problem is getting in. . . you need to make reservations quite a ways in advance, as Moab is a very popular place after about May 1. We were last there three years ago, and feel lucky to have landed a spot this year.

Zion - There's three good campgrounds somewhat close to Zion. We stayed at Zion River RV Resort near Virgin, UT a few years ago. Ten minutes (at most) to the park. Nice RV park. We have also stayed twice (with a third time booked) at Sand Hollow State Park a few miles outside of St. George. Maybe 20 to 30 minutes to Zion. Very nice state park, full hook ups, locked down at night, etc. Stay in the West Side Loop. While we have not stayed at Willow Winds in Hurricane, UT, we have driven through it, and it looks like a nice park. A friend of ours stayed there for a month or so and liked it. Maybe 20 minutes to Zion NP.

Boise - We have stayed many times at the Ambassador RV Park in Caldwell, ID, just west of Boise. Easy on and off the freeway, near a Pilot station, quiet, and nice wide sites.

Salt Lake City - We've only stayed once at the south end, and would not do so again. However, we have stayed at the Pony Express RV Park at the north end (North Salt Lake) where I-15 and I-215 merge, and have found it to be a nice place to stay. Nice long and wide sites, easy access to SLC. Pilot station nearby. RV dealer next door, with a nice parts store. Has a growing number of permanent residents, but so far that has not been a problem.

If you are just looking to stop for a night, this should work:

Fallon Churchill Fairgrounds
325 Sheckler Rd
Fallon, Nevada 89406
Phone: (775) 423-7733
Not a spectacular place, but has the basics. Easy access.

~ Santa Fe, NM
~ Taos, NM

Santa Fe: First choice: Santa Fe Skies. Santa Fe Skies Web site
Second, Ranchos de Santa Fe.

Taos: Taos Valley RV Park https://taosrv.com/

http://www.emnrd.state.nm.us/SPD/FindaPark.html

Reservations IMO “lock” you in. I prefer discovering as I go and camping wherever when necessary. As a new RVer, learn to use your on board systems. They work well. Fresh water is usually the limiting factor.