I would suggest staying in West Yellowstone. We stay at Buffalo Crossing. It is located within a couple blocks of west entrance. There is many things to do in West Yellowstone.
We have been to Yellowstone many times. We figure a long day to do each of the loops. As a new visitor figure at least two days per loop without spending much time on the trails. Double the two days to do some of the trails.
Fishing Bridge is concessionaire operated. It is not cheap any more. It is crowded with rigs close together.
Madison Campground in western portion is nice. There is no electric. Water and dump station is available.
Jackson can be seen coming or going by leaving through south entrance. Jackson itself is a couple to a few hours. Teton Park is a good day for visiting depending on your interests.
You could go to the NPS.gov website and get a lot of information on Yellowstone.