South Dakota has a lot of excellent climbing, but most is concentrated out west in the Black Hills.
The southern part of the Hills boasts home to the Needles, one of the last bastions of traditional runout face climbing in the U.S., as well as the granite sport routes near Mount Rushmore National Memorial. Lots of first ascent possiblities remain if you are willing to walk a few miles.
The northern part of the hills near Spearfish is laced with limestone canyons, many of which have been developed as sport areas. An enormous amount of new route potential remains; there are many undeveloped limestone canyons with good steep rock and numerous granite formations with no developed routes throughout the Hills.
The Black Hills are ringed by sandstone, with some good developed and undeveloped bouldering with an odd route or two. There is good stuff right inside Rapid City.
In the winter, ice forms in both the Needles and Spearfish Canyon. These are some of the nicest moderate ice routes anywhere, and not as crowded as climbs of similar quality in Colorado.
In the eastern part of the state Pallisades State Park northeast of Sioux Falls contains a fair number of quartzite cliffs. Not spectacular compared to the granite and limestone in the Black Hills, but out there on the flat you take what you can get.