Where I would recommend, depends on what amenities you want.
If you want full hookups, then make reservations at Fishing Bridge.
Camping in the park, I was last there in 2006 and stayed at Fishing Bridge. It was basically like an RV Park, not a "campground" with spaces between sites, etc.
Where you stay in the park also depends on the size of your rig.
I did a drive through tour last fall and the few campgrounds that were open in late September, weren't what I'd consider big enough for anything over a 20' trailer, except for Mammoth Hot Springs, but that campground has very limited shade.
West Yellowstone has a few parks. The two KOAs are close to town, but cost a fortune during peak season. The one further east is the nicer of the two.