The most quoted figure is 300 miles/day for comfortable traveling, plus or minus depending on the locations of stop points. You sound like you want to do a leisurely trip across the southern tier of states to Vegas and then back thru South Dakota (Mt Rushmore). Cutting up thru Utah from Vegas you will be passing the major Utah parks: Zion, Bryce Canyon, Arches and Capitol Reef so you may want to at least do drive thrus of those parks.
Our usual tactic is to pick out a central point for an area we are interested in. Camp there and do day trips out from there. For example we camped in Cortez CO for a week and did trips out to Mesa Verde, Hovenweep, Canyon of the Ancients, Valley of the Gods and Monument Valley from that location. You have that little A frame so moving from place to place may be less of a hassle and you may want to move more.
So I suggest making flexible plans since you may want to spend more or less time in any one spot. Reservations are mainly needed at major National Park Campgrounds and the days around a major holiday, 4th of July for example. Otherwise we just make the reservation when we are 2 or 3 hours out from the CG. We rarely have a problem.