we've been to the bighorns a number of times. some favorite camping spots for us were:
west tensleep lake (north off of hwy 16 on forest road 27) the turnoff is just past meadowlark lake) it is a nice small campground with some sites right on the lake.
tie flume or dead swede (east off of hwy 14 on forest road 26) have seen moose and mule deer in that area.
medicine lodge historical site has a campground (northeast of hyattville off of hwy 31) a nice little campground run by the state of Wyoming also includes some sites along the creek. a shortcut to it from hwy 16 is to turn north on nowood rd just west of tensleep.
don't miss a stop of a night or two at devils tower nm. nice campground inside the park. belle fourche cg
we also camped at big Sioux recreation area near Brandon sd (just east of Sioux falls sd) on the way out and back. very nice and is run as part of sd state park system.
wall drug and the badlands. wind cave or jewel cave in the black hills. mount rushmore. circle b cowboy cookout is the best one in the black hills (we've been to the others, too)