We did something like that except I picked up my wife at Bozeman airport some thing you may want to consider as the point for her to fly out of unless you want to go to SLC.
As for the Black Hills there are basically these main spots to see:
Rushmore--no more than a half day
There are two caves- a half day at each
Custer State Park-- A day or so and it's the prime place to stay
Badlands NP- Can just be a drive thru plus stopping at various pull outs or longer if you are going to do the trails. Wall drug is in the same area for a hour or two of tourist glitze.
There are other sites but I'd skip them if you want to maximize your time at YNP. There are many private CG in the Badlands.
Think about the short detour to see Devil's Tower on the way to YNP.
There is no best way to see YNP. We have spent maybe 6 weeks there over the years and still haven't seen it all. But concentrate on the geyser basins, spend a day at each of them. Then there are a lot of other places to see if you have time. A day trip to Tetons is something to think about.
Let me warn you if you don't have reservations by now you could have a problem. The main CG in YNP--Fishing Bridge will be closed for renovations.
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