I did this exact trip from Dayton, OH this summer. We stayed at Custer State Park and from there visited Wind Cave National Park, Mt Rushmore, The Badlands, Minutemam Missile Site, Mammoth Site, Needles Highway, Devils Tower and a few other odds and ends. From there we headed west to Yellowstone. Take 16 through the southern portion of the Bighorn Mountains to RT 20 north and then hop back on 14 to Cody. There are a couple decent pulls but nothing too crazy. You'll see more RVs on that road than you've ever seen in Ohio. Once you get to Cody you are an hour from the gate to the park, and once you hit the gate you are another hour to the loop. I recommend staying at Fishing Bridge in the park. It's basically a glorified parking lot in the woods but it's the only campground with full hookups in the park. After a solid 4 days or so in Yellowstone drop on down to The Tetons for a couple days and stay at Colter Bay in the park. The trip back to Ohio from there starts taking my you southeast though Wyoming through the Wind River Indian Reservation until you run into I-80 and head East.
I forgot to mention...... as far as weather goes it's a complete **** shoot. We had weather in the upper 80s in SD, and it snowed on us in Yellowstone..... On July, 11.
