I would suggest you make your reservations to get a spot, being you are coming from so far. Do you want hookups or not. We stayed in Custer State Park but we did not have hookups at the time. Rafter J in Hill City has hookups and is a nice park. Someone also mentioned Sylvan Lake which is beautiful.
My brother suggested we see the Cosmos. I am glad we did it was really amazing. My DH is an engineer so it was fun to see his expressions. DD also enjoyed it also. http://cosmosmysteryarea.com/
We love the Mt Rushmore area. As far as Mt Rushmore, try to get there before sundown. They do an amazing laser show with patriotic music. They have plenty of areas to park so that is not an issue. You pay for parking but entrance to the monument is free.
Enjoy your road trip. Ours is over for this summer, school starts in 2 weeks.