It's hard to say when is best being as how you didn't say what you wanted to see or do. I personally like all of the seasons, and find there is a lot to do here year round which is a big part of why I settled here when I retired from the Air Force.
If you're coming to ride the Paiute atv trail system, some of the best higher elevations portions of the system don't become rideable until July most years. In the lower elevations it can run somewhere between 100 and 110 degress that month which is great for boating at lake Powell, not so not so nice if you're climbing Angles landing at Zion National Park. Even then, you can have daytime timps in the 70's at Cedar breaks National Monument and temps around 105 at Zion less than 50 miles away as the crow flies.
My own preferences based on region. Arches/Canyonlands area, September thu April. You might get some snow Dec-Feb in particular but it tends to melt off quickly and the area is fantastically beautiful then. Expect cold nights and reasonable daytime temps so you'll want an RV capable of dealing comfortably with the colder nights. Summer is also nice, but hot. Bryce canyon area I like year round. Zion NP definitely prefer Sept thru April. Flaming gorge and even down low in the vernal area is a great summer area as are the Uintah mountains Henry mountains, Abajo's La Sals etc. I could make better recommendations if I knew whether you were coming to ski, sit around a campground, hike, climb, boat etc.