When we went to Mount Rushmore, we got up and went VERY early. When we got there we parked where a couple other cars were. There was no one to take any kind of admission.
We walked in, walked around, took pictures and saw no one until we were ready to leave. Then we say a guy hosing down the walks, so we stopped and talked to him a bit.
Custer State Park, Bad Lands, Devil Tower, Keystone, and Spearfish Canyon are at the top off the list, but there are so many things to see and do, I hope you have plenty of time.
If you are going from the east, a stop at the Corn Palace is interesting, but I wouldn't make a special trip. It was just on the way.