Offroading ideas Rapid City SD, Red Lodge MT, Yellowstone/Grand Tetons, Driggs ID

If you have the time while in the Black Hills and Custer State Park do take the time to drive the full loop - not offroading but the roads have plenty of scenery and twists and turns to keep it interesting. Iron Mountain road and the Needles Highway.
On your way across Wyoming plan to catch Highway 14 over the Bighorn Mountains if you can. Beautiful area. Steep drive though. Not recommended if you're towing a trailer....that kind of terrain is not for the faint of heart. But neither is the Beartooth.
Red Lodge is a cool little town, lots of shops and restaurants along the mian street. I recommend the candy store.
Going over the Beartooth watch for a sign on the right while you're still below the tree line that says "campgrounds". That leads to the Parkside/Limber Pine CGs and just past the CG's the roads to Helllroaring Plateau and Glacier Lake Trailhead. Little rough, kinda steep. Probably going to have a ton of campers and ATVs to dodge but its a pretty area to explore and definitely worth the time.
After that you are pretty much committed to the Beartooth Highway, not a lot of side roads since its a wilderness area.
If you have the time to drive the Chief Joseph Pass from near the top of the Beartooth down to Cody I highly recommend it as well.
Closer to Cooke City watch out for the signs for Lulu and Daisy passes just past the turn off to Colter Campground. Dirt roads that make a loop just to the north of the highway. There are lots of other roads that tie in back here so be sure to have a GPS and map, not a lot of signage once youre on the dirt roads.
Outside of Yellowstone on the West side I recommend taking the detour off of Highway 20 to the Mesa Falls Scenic Byway. Takes a little longer to drive but the scenery is great and the Mesa Falls Site is worth a stop.