It's a huge park and hard to see in one trip which is why we've been back over a dozen times, but we live close enough that we can do that. For starters, the best entrances for RV parks with hookups would be the West Yellowstone entrance, the South entrance, and the North entrance. At the West Yellowstone entrance are several RV parks in West Yellowstone, including a FS CG with electric sites and water refill stations. At the South entrance, the best places really would be either Colter Bay RV or Flagg Ranch. Anything in Jackson is going to be side by side type of RV parks with little room to breathe. The North entrance, there are some RV parks in Gardiner.
We've stayed multiple times in West Yellowstone at Baker's Hole FS CG at both electric and dry sites, at Grizzly RV, at the West Yellowstone KOA, and as far out as Red Rock RV, past Henry's Lake. Our personal favorite is the FS CG, then Grizzly RV, with the KOA being ok, clean. We've also stayed to the South at Flagg Ranch and Colter Bay. We also have stayed to the East over the Togwotee Pass at Falls CG that has a loop with electricity and they too have water. This one is far out though, but we prefer that and enjoy the drive. We've never stayed in Gardiner and have stayed along the route to Cody, but focused on the Cody area and Chief Joseph Pass.
We've always gone in September and find that it's a good time to go. In early September most services are open, later they are not, but you can still find something. We've stayed in the park at Bridge Bay, but it's dry camping. In September, there are few campers. We've never stayed at Fishing Bridge due to the tightness of the sites.
If you plan to stay at any of the RV parks, reservations are recommended for Colter Bay and Flagg Ranch as well as the others, but we've managed to get in without, especially to Grizzly RV.
2011 Bighorn 3055RL, 2011 F350 DRW 6.7L 4x4 Diesel Lariat and Hensley TrailerSaver BD3, 1992 Jeep ZJ and 1978 Coleman Concord Pop-Up for remote camping
Dave & Renee plus (Champ, Molly, Paris, Missy, and Maggie in spirit), Mica, Mabel, and Melton