Where to stay depends on whether or not you need hookups. If so, Fishing Bridge is the only campground in the park.
If you don't need hookups and are staying 7 to 9 days, then staying in more than one campground might be beneficial. (Madison and Grant Village would work.)
If you go in early June, do not consider Glacier National Park - Going to the Sun Road would probably not be completely open. This year, which was a low snow year, it opened around June 19. In 2014, it opened on July 3. (I wouldn't do both Yellowstone and Glacier unless I more than 9 days. We spend 8 days in Glacier and 8 days in Yellowstone/Tetons this passed summer.)
Instead of Glacier, consider going south to the Tetons. I believe the Yellowstone entry fee will cover the Tetons, as well. (Since we use my senior pass, we don't keep up with such things :)) In the Tetons, Coulter Bay and Flagg Ranch have hookups. Without hookups, I would choose Jenny Lake of Signal Mountain.
Have fun.
Tom