its a pretty good drive from Yellowstone to the Tetons (Jackson Wy), both distance, and very slow road inside Yellowstone. VERY SLOW, lots of gawking at buffalos on the road.
I visited Yellowstone mid-June, snow was deep, air was cold. South entrance just opened, many lakes still frozen, most side-roads still closed.
It's one of those areas that have a very narrow "prime" window to visit in, it's gonna be crowded - take it easy while you're there, if possible make reservations NOW.
The Badlands was great. Lots of folks say it's boring, just a drive through, just a day trip, I dunno, I find the colors in the sandstone hills amazing - could have easily spent a week there watching the colors from the various overlooks, driving the more mild 4wd trails. The colors are amazing especially at sunrise / sunset. There's a tiny NP campground inside the park, some sites have electric, some don't.
The Black Hills are another gem. Lots of campground options, LOTS of neat lakes and trails to see, Needles Hwy is a fun drive. Rushmore personally was a disappointment, it looks cooler on TV imho
