Oh! There is so much to see in this part of the country it is a little like asking what kind of car you should you buy.
The Black Hills should, of course, be included in the trip you suggest. Mt. Rushmore will be a disappointment (the crowds and overwhelming commercialization heads the list of many reasons why) but don't fail to stop anyway. On the other hand, Custer State Park will be something to remember the rest of your life. Also be sure to stop at Devils Tower to get Spiritually "centered" on the way to the Big Horns. (If you can take a side trip to the Little Bighorn/Greasy Grass Battleground you won't regret it.) From Buffalo (Wyoming) head out on Highway 16 to Worland and go through Thermopolis to Riverton and then head over to the Tetons and Yellowstone Park. This is one of the most visually impressive drives in the US.
From Yellowstone the decision-making becomes more difficult. The drive to Coeur D'alene (many routes) would be at the top of my list except that you end up so far from the Grand Canyon area. With that in mind, go from Yellowstone south through Idaho Falls and Pocatello and Salt Lake City to I-70. From there, any road you take south from there will be the best choice.
I didn't address the getting to the Black Hills but it is just as wonderful as what I described above. For example, I spent a week earlier this year on Old Route 66 between St. Louis and Chicago and can recommend that as a destination.
We have put around 20,000 miles on our RV since July and have never made a reservation earlier than three hours before arrival. I suspect that most campgrounds will have even more vacancies this year than last. In any event, we have never had trouble finding a campground that didn't have a place for us -- we have never spent more than one night at any one CG.
If you are looking to save money on your trip, you should look into Passport America, the 50% discount pays for itself very rapidly. We have also found the Good Sam club to be very useful also -- if you can put up with their junk mail campaign. However, for a consistently good experience at a campground, I highly recommend using the KOA network -- be careful, however, because they have a lot of very tempting extras. (In fact, there are those who believe KOA stands for "Keep On Adding.")
I have no advice on the generator. Our RV is self-contained and the vehicle's engine keeps the house batteries charged up. Nevertheless, I would feel very vulnerable without the propane tank and/or the generator. BTW, if your house batteries are charged up, the only reason you need a generator is the air conditioner, microwave, coffee maker and such electricity hogs -- things a true "camper" (not me) can easily do without.
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"It's tough to make predictions, especially when it involves the future." ~Attributed to many
"In theory, there is no difference between theory and practice. But, in practice, there is." ~(perhaps) Yogi Berra
"Those who have knowledge, don't predict. Those who predict, don't have knowledge."~ Lau tzu
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