I'm at Mt. Rushmore now. Came up from TX. You can also take I-80 from the Chicago area to avoid some of the high winds that Lolead talked about (and I'll attest to, as well). You can then take either 83 & 91 out of North Platte, NE (as I did) through the Nebraska Sand Hills up to I-90 (about 1/2 way across SD). You'll miss the Corn Palace, but you can still catch Wall Drug, The Badlands, and some other sites on your way to Rapid City. Or you can stay on I-80 to Ogallala, NE, then take U.S.26 to U.S.385 and take that to Hot Springs, SD. From there, you have your choice of 3 different routes up to the Rapid City area. Which one depends on what you want to see on your way up.
I'm currently staying at a Good Sam campground just south of Rapid City, called Happy Holidays Resort. They have full hook-up sites, a clean laundry room ($2 wash/$0.25 for 13 minutes dry time), campground store, propane and gas/diesel, game room, playground, and pool. It may not be as plush as some of the KOA's or resorts, but it's 1/2 the price of those places. It's also $10 less than the only campground in Custer State Park with electric hookups. There's plenty to see within a 10 minute drive; The Faces are about 20 miles away, and both Crazy Horse & Custer SP are about 25-30 miles away.
Whichever way you go and where ever you stay, it's a great place to visit. Hope you enjoy your trip as much as I have.