John, I'm pretty much going on memory of the state, having been to most of its sights and corners over the past 50 odd years. Parents grew up on either side of the James River (runs from North Dakota southward through SD until spilling into Missouri River) south of Mitchell, SD and I've seen a fair amount of that SE region.....a bit uneventful to me. Were I to do such a trip, I would start up in the Coteau around Webster. That is an unusual region where the glaciers stopped and the NW-angling edge of the Coteau represents the SW 'beach' of old glacial Lake Agassiz, the bed of which makes up the Red River Valley roughly from Sisseton, SD up through Winnipeg, Manitoba. From there, you may be able to find scenic routes along the James River to get to either Hwy 212 or Hwy 14, both east-west routes. Not saying you should take these as they are highways, but there should be state highways that parallel these over to the Missouri River near Pierre. From there, Hwy 14 and state hwy 34 become the same and it meanders west to the well-known burg of Sturgis. From there, you are a stone's throw to the scenic Black Hills and a trip through Spearfish canyon would be worth taking in. Not sure how you plan to end the trip, but some cruising through the Black Hills to end up at Hot Springs or Edgemont would provide a lot of scenery.
Again, just some possible notable routes with good scenery in case it helps.....just don't know much about specific roads that are recommended for bicycling. It may also be worth polling others about an east->west ride vs. west->east as I would think the prevailing winds from the west may pose formidable travelling by bicycle, but just guessing here. Good luck!
Source of map photo:
https://cdplodge.com/wp-content/uploads/2012/09/Coteau-des-Prairies-Color-Elevation-Map.jpg