Custer State Park, the Black Hills, South Dakota

Skip Mount Rushmore, and go here instead.
We hadn't been to Mount Rushmore in almost 30 years, and the intervening years have not been good to it.  There is an elaborate Visitor's Center and loads of people.
This is it, the beginning of the west, right there on the edge of South Dakota.  From here you go on to see the Big Horn Mountains and the Wind River Range and all the wonder that is Yellowstone.  But a stop here is well worth it, and amazingly, my oldest son lives within an easy day's drive of here (for us, it is a more substantial day, but still doable).
The mountains are not the Rockies, but they are worthy of the name, the lakes are beautiful, and best of all there is a lot of animals about.  The hallmark is bison, but I saw Dahl sheep, which I really love.
We had a very short visit, but it was a place that would be well worth returning to.