Do not miss driving the Wildlife Loop at Custer State Park! As a native of South Dakota, I've spent plenty of time in the beautiful Black Hills area, and this is by far the best place to see wildlife there. Everyone does the Needles Highway drive, but many people miss the treasure that is the Wildlife Loop Scenic Byway (because of this, there are usually not too many people on the road and it makes for a leisurely drive). Eighteen miles of beautiful views of the forest and mountains, with several interesting stopping points along the way, and more wildlife than I've seen in the rest of the Black Hills combined. We always see some of their herd of 1300 bison towards the end of the drive, and they're usually very close to the car, making for excellent photos (though don't disregard the signs about bison being dangerous--they are--and stay at a safe distance). We also always see white tail deer, mule deer, antelope, various birds, burros that come up to your car seeking a handout (they're not shy about sticking their head into your car window if you let them either!), and various other creatures. Don't pass by the opportunity!
Mount Rushmore is definitely a unique place. Originally designed as a tourist destination, the state of South Dakota was looking for something to drum up visitors. That's when the suggestion of carving out a monument to some of our past Presidents came up, and the rest is history. Excavation took around 14 years and now we have the Mount Rushmore National Memorial. You can get close to the heads of Jefferson, Washington, Lincoln, and Roosevelt (Teddy), as well as learn about the men who carved the mountain and how it has taken part in history since it's dedication.
The area is steeped in controversy too, with lingering questions over the designer, the inclusion of Roosevelt (some think he's not worthy, but he's one of my favorite prez), and the Native American presence in the area before, during, and after the carving. It is a tumultuous place to visit, but no matter what your beef is, you have to admire the sheer effort that went into creating this space. It is strange to see in person, after having seen it so many times in other media, to have it right in your face.
Wind Cave National Park is just a little jaunt down the road from Custer, and holds in it some amazingly beautiful caves. That is really all there is to do in the park, there are few hiking trails, the big draw is the cave. There are several tours, taking you deep into the bowels of the Black Hills.
The caves there are wonderful, beautiful in their own right. They do not have the dramatic stalactites and stalagmites of traditional caves. Instead, there is delicate formations known as boxwork, the result of water wearing away limestone while leaving the impurities in the cracks. It is gorgeous. The whole cave is wonderful, a delightful day trip south of Custer SP.