Years ago, I worked in Yellowstone NP and stayed in West Yellowstone, Gardiner, and Canyon Village. Cody and Colter Bay (Teton NP) were too far from our worksite to commute on a daily basis. As others have said, Fishing Bridge is the only campground in the park with electricity, but, if your RV is less than about 30 ft. and self-contained, you can stay in nearly every gorgeous campground in the park.
Since Yellowstone is the size of many Eastern states, I personally would not just pick one campground and drive around the park from there. I would move two or three times to be able to experience Yellowstone Lake and Falls and the animals in Hayden Valley from one campsite, the Firehole and Madison Rivers and Old Faithful from another, and Mammoth, Tower Falls, and Norris Geyser Basin from another.
And I would also stay at the Forest Service campground in Cooke City. The drive over the Beartooth Highway from Cooke City to Red Lodge has been labeled "the most scenic drive in America." I highly recommend that you try it WITHOUT YOUR RV (it's a toad or tow vehicle ONLY road!). You can make it a loop drive from Red Lodge by returning on MT 72 and WY 296, so you don't have to drive the Beartooth both directions.
No matter how many times I have been to Yellowstone, there is always a new surprise on each trip. Like a herd of buffalo walking right past our car early one morning when we were the first car to arrive in the Yellowstone Falls parking lot (no, we did not get out until they moved a LONG way away). Or stopping to photograph a family of trumpeter swans floating serenely down a glassy-surfaced Madison River. Or seeing a geyser erupt in Norris Geyser Basin (the hottest in the park) that only erupts about once a month. So, wherever you decide to camp, it will be a great trip!