I made a trip to the Black Hills last August, I also got to watch the eclipse from Alliance, Nebraska, after touring the Black Hills, I was gone from home in Georgia for 12 days and did not get to see near enough in the Black Hills, it was more of a scouting expedition for the next trip. The Black Hills is a trip unto itself, and you need to allow 14 to 16 days from SC for such a trip.
I have been planning a "four corners" trip to cover the Painted Desert/Petrified Forest, Grand Canyon, Zion, Bryce, and a host of other lesser stops along the way, and quickly realized it would take two trips to cover everything I wanted to see.
Other issues, there are six of you, what kind of "RV" are you using, and what is the physical condition of the persons in your party. If your inlaws are older and have any mobility issues at all, this will slow down things greatly. Also allow for a "day off" from the touring to stock up on groceries, take a breather, relax, and just enjoy a view.
Many campgrounds out west, from everything I have read, do not have any hook ups, at least the Federal campgrounds, and some of the state ones. You need to be prepared with a generator and full tanks of water. Also if your RV is large (such as a 40 ft MH or even worse, a large 5th wheel and its tow vehicle) you will be limited as many of the public owned campgrounds won't have large sites. A trip like this takes immense planning. You will want the best tires possible on the RV. If a trailer you will want two spares, jacks and tools for changing them, as cell service may be spotty and roadside assistance may not be able to help you.
Your proposed trip is better broken up into a dedicated Black Hills trip, an "I-40" trip to cover the sights of New Mexico and Airzona, and lastly a Utah trip to see some of what it has to offer, and you still haven't seen the Rockies yet!!!
I know I keep plugging my thread on my
Black Hills, Badlands, and Eclipse trip, but it is certainly worth a read here so as to give you an idea of the time frame involved in a trip like this.
EDIT: After re reading your original post, you state you have never RV'ed before. Are you planning on renting a Class C motorhome such as one from CruiseAmerica or ElMonte? if so, be aware these are usually on the smallish side and sleeping for six will be crowded. If you are renting, you are best to fly out west and then pick up a MH there.
If you are simply new to RV'ing and have bought something, relax, take a few local trips up into the smokies, the Blue Ridge Parkway and other eastern sights and get acquainted with the RV and the ways of RV'ing. Otherwise it could turn out like Robin Williams in the movie RV.
https://youtu.be/foiHTT2rcso
Charles