All those suggestions are good. The Black Hills is the best part of SD IMO. If you have time, check out Deerfield. Very nice. Sheridan Lake is quite pretty and close to Rapid City. Terry Peak and Harney Peak are neat if you want to make the hike. IIRC the latter is the highest point in the US east of the Rockies.

The Badlands are very pretty. Best go later in the season when it isn't so hot. The really neat thing about SD is that in about 2 hours you can drive from flat out desert (Badlands) through grass/steplands, across open prairie, through river break country and right up into the mountains. Many different micro-climates and types of terrain.

If you head as far as the middle of the state, the Dam is pretty cool to see.

As Hikermor said the caves are a lot of fun. Beyond that it's dependent on your view of the tourist traps. They can be neat in a hokey way.

I was born in SD and have lived here all my 44 years. For the last almost 20 years I've lived in Sioux Falls, in the far eastern part of the state. Most of the people are in the east but the prettiest scenery is in the west.
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“I'd rather have questions that cannot be answered than answers that can't be questioned.” —Richard Feynman