Visiting Yellowstone you have a number of options, much depending on the time of year.
Spring is the time to see wild life which is primarily in and around the Lamar Valley on the East side of the park. If you are keen for this, you will want to focus your trip in areas like Cooke City and surrounding areas for camping.
Summer (late June-early September)
Let's call a spade a spade, it's going to be busy no matter where you go. I always say that the best place to stay is outside the park (unless you got a camping spot inside the park) - just don't feel bad if you can't get a spot inside the park. The plus with Blacksford and your season pass, there are usually express lanes for passes and there is something to be said about early birds getting the worm as well. Also, knowing that it's busy, relax and enjoy the ride. Pop the windows down, enjoy the pull offs, go for a short hike and park in the back of the parking area (it's easier to get in out with the RV's).
Fall (September-October onwards)
The sweet time in the park. Kids are off to school, campgrounds are easier to get spots. It's cool evenings and warm days. Animals are coming down from the mountains. NOTE: it's not unusual to see snow in the fall starting in September and October in Yellowstone.
You have a couple options for camping around Yellowstone and Teton:
1. Reserve Camp Site in the park here: https://www.nps.gov/yell/planyourvisit/campgrounds.htm - these tend to fill up quickly.
2. In West Yellowstone: Grizzly RV Park, KOA West Yellowstone
3. In Jackson: http://jacksonholecampground.com/
On the east side of the park you have Gardiner or Red Lodge:
1. Yellowstone RV Park: https://www.rvparkyellowstone.com/ - same owners at Grizzly in West, on the river
2. Rocky Mountain RV Park: https://www.rockymountainrvpark.com/
3. Perry's RV Park, Red Lodge: https://perrysrv.us
Ideally, we recommend you pick a spot to stay for a few days, move on to the next. A few good places to camp on the night before you come back to the airport are:
1. Norris Hot Springs - https://www.norrishotsprings.com/ - mention us and you get to soak for free. Great food and experience, a locals favorite.
2. Red Mountain Campground https://www.blm.gov/visit/red-mountain-campground - on the Madison River, beautiful!
3. Moose Creek Flat Campground: https://www.fs.usda.gov/recarea/custergallatin/recreation/fishing/recarea/?recid=5605&actid=43 - on the Gallatin River, stunning and great fishing!