We visited Mt Rushmore in June. We stayed in a hotel in Custer, it was much cheaper and we enjoyed it more than the town of Keystone (which seemed a little cheesey to us).
We absolutely loved Custer State Park. Maybe even more than Mt Rushmore. Ask the park rangers where the buffalo have been spotted when you enter the park. We found them right where the park ranger said they would be, it seemed like they made their way down a hill, across the road and to a stream every morning, then back up in the evenings. We managed to catch both times on two different days and the free-roaming buffalo were right next to the car. I could have reached out the window to pet them if I wanted (not recommended by the way). We also saw lots of other wildlife in Custer State park (deer, antelope, etc). All the wildlife was on the southern half of the park (at least when we were there). We also drove the northern loop, which is a spectacular highway. From the main road that runs east-west through the park take the eastern-most road to the north, which takes you essentially to Mt Rushmore. The drive is not-to-be-missed, it is spectacular. There are some very, very cool tunnels and some very fantastic views of Mt Rushmore driving it this direction.
The other not-to-be-missed attraction (especially with kids) is Storybook Island in Rapid City, SD. It is free admission, set up and maintained by the Rapid City Rotary Clubs. It is sort of like a giant park with lots of different little cottages set up, each one showing a different mother goose story. For example there is Old MacDonald's Farm with a petting zoo, The 3 little pig's house with a pig chute (slide), the Cat in the Hat's house, dragons and a castle, a leprechaun with a pot of gold at the end of the rainbow, Jack's beanstalk to climb, Yogi Bear, etc. It is fun for adults to see and a great place to run off some steam for kids. Plan to spend at least 2 hours there, you could easily take a picnic and spend an afternoon.
I also heard wonderful things about the geology museum at the South Dakota School of Mines in Rapid City. Check their website for hours before you go. Unfortunately we never made it there when they were open.
We also visited Jewel Cave National Park, it was neat for me to see, unfortunately my kids didn't enjoy it. Maybe yours would be more into it, I did see some kids there having a great time.
My only other tip is to be careful driving, especially at dusk. We were driving from Denver into Custer and it was about 9:00pm, we were about 10 minutes outside Custer when the hugest elk I've ever seen ran right in front of the car. After that I slowed down and we saw at least 3 more standing just off the side of the road.
If you have any other questions or want to see any of our pictures, feel free to email me. Enjoy your trip!