Leaving Sturgis and riding straight to Yellowstone......I've done it a couple times. Let me say that on the eastern side of Yellowstone is Cody, Wyoming that is a town full of history thanks to Buffalo Bill.
If time allows (it's a days ride from Sturgis) spend the night there in Cody. The next morning, get up and go to the museum..........it's incredable if you're interested in western frontier history. I did a quick tour in about hour and a half or two hours but it was well worth it.
Then again, if time allows, do a complete loop around Yellowstone before heading south to Jackson. In Yellowstone be prepared around every corner for wildlife in the road. The moose will usually move pretty readily but a heard of buffalo will stand there as long as there's grass on booth sides of the road. The coyotes, wolf and bear usually run off as soon as a vehicle stops for better viewing. But you'll definatley see mother nature at work there! Enjoy......