We stayed in Jackson when in the area. The Tetons were in easy range and Yellowstone if only driving thru is relatively easy to get to from there.
Also in Jackson the National Elk Refuge is there and has a great visitor center and you can spend some time there plus wandering around the town is nice.
With a dually or long pickup most of the parking spaces will be confined so be forewarned.
As to the black hills guess it depends on what you want to see. Rushmore is a good couple of hours of looking walking around, crazy horse is a tourist trap, went there a couple of years ago and it has not really progressed from the 1st time we went there 20 years ago. Supposedly the black hills are a good motorcycle trip but mainly set for tourists as far as I know. Have friend thats been to the rally several times and he never really talks about much more than the rally and all the T&A that is there...