Agree that you'll want to be at Mt. Rushmore pretty early in the afternoon. I was there for the Fourth of July a couple of years ago, and it was very crowded. I ended up parking along the curvy road between Mt. Rushmore and Custer State Park. There are some pullouts along that road with great views of Mt. Rushmore. A lot of people do this as well, and picnic while they wait for the fireworks.
I highly recommend the scenic loops in Custer State Park for watching bison and checking out the prairie dogs.
Also, be careful with your speed. When I was there, there were a lot of police officers out and about looking for speeders. I got pulled over while rushing to drop off my rental car in Rapid City. They stopped me about 1 block from the rental place, and let me off with a warning.