I have been going there since 1955. I like late May/early June to the see the babies and Sept 19 for the elk rut. Bad weather can occur any time so be ready for it on the Plateau.
Hook ups are hard to find in the park. Most people with big rigs commute from a place like W Yellowstone (west entrance). I have alwasy liked the east entrance because it is great moose habitat. Mammoth is a good place to stay but hard to find hookups.
The real secret to visitng the Park except for avoiding July and August, is to get out a walk. Ninety percent of the visitation is within 1/4 mile of a road. There are some great dirt roads in the park. Take those and get out and walk.