You will be in my neck of the woods! In terms of friendly activities, most national parks won't let the pups out of parking areas
I am not sure about hiking trails, etc near there, but I would guess they would be a good bet. I haven't ever been to that area with a dog. However, I know this is not where you are going, but there are places in MN that are very dog friendly. Tons of restaurants and breweries in the Twin cities are dog friendly, along with lots of hiking trails and parks. We love going up to the North Shore with Bacon. The superior hiking trails are amazing and dog friendly. I can talk to some relatives we have over in SD and get back to you, they have a labbie and should know some places to go!
ETA: I know the Crazy horse memorial is very dog friendly, so if you are interested you could do that!