When we went, we decided not to stay in Yellowstone itself because we feared it would be pretty crowded (it was). We camped at Henry's Lake State Park across the border in Idaho - about 20 min away. For us it worked great. We drove in to the Park every day in our car, used our car on the narrow roads, parked at the crowded exhibits where cars park, and generally had a great time.
One tip: When you arrive, make your first stop to the visitor center. Not only is there a great film there, you can talk with the rangers. They know which attractions are working at the time you're there so you won't drive a long ways (and the distances are large) only to see a dribble. They also know where you're likely to see wild life.
We enjoyed eating at the lodge - once - just to say we had done it.
Hiking is great... no, fantastic, there! Comfortable shoes is a must.
We did the Grand Tetons in a second trip. They are magnificent. So nice, we renewed our spot twice so we could spend extra time! Oh, a tip for the Tetons: there are multiple visitor centers. They are NOT all the same. Each has its own specialty and ALL of them are absolutely worth taking the time to visit & talk with the rangers at each. The movies are different and well done. For that trip, we did camp right in the park at Colter Bay. We would also have liked to camp at Jenny's Lake (we did drive through it and liked it - more wooded and rustic than Colter Bay) but it was full up while we were there.
Enjoy your trip. You'll see some of the best that the West has to offer. On your next trip, check out Southern Utah.. the Grand Canyon.. and Mesa Verde .. then North to Glacier.