Here's a couple old posts, maybe something will help in your plans:
Wyoming Area Black Hills Area Not sure you plan on taking in the Tetons or not. We stayed at Colter Bay Campground, the one without hook-ups in mid-June 2010, it was a nice place for such a large campground and close to showers. Our daughter & her hubby needed them, not having showers in their A-Liner. I have heard good things about Gros Ventre Campground...get more animals walking by or close by, etc. Nice views.
If there again, I believe I'd try Gros Ventre.
The Colter Bay Visitor Center and Indian ArtsMuseum by the Colter Bay Campground is really a neat one.
On our 2nd trip to Yellowstone, we went up to Mt. Rushmore & Black Hills, then went to Yellowstone via Cody, WY. Stopped by Devil's Tower & detoured up to see the Little Big Horn Battlefield on the way to Cody.
Devil's Tower But entering Yellowstone from the North Entrance might work as well using I-90 to Livingston, MT. Then to South to Gardner & the Entry Gate on up to Mammoth. We enter that route in 2009. Not as much climbing as via Cody.
However, IMHO, One might want to go to Yellowstone/Tetons first, then if time allows head over to the Black Hills.
We've never entered by way of ID. Can't help with that. We've always wanted to be in the park when seeing it. Have never had a reservation, always found a site.
:CDesertHawk- Las Cruces, NM USA
*2015 Lance 1985~Casita de Campo~23' 4" Tongue to Bumper, Dinette Slide
160 watt Solar Panel/GoPower! Solar Controller
*2009 White Ford F-150 Reg. Cab
Long Bed with A.R.E. Molded Fiberglass Topper
*Previously~ 2005 16' Scamp