We went to yellowstone last year. We went in october but if you're going in the summer, you may want to consider staying in the park since thats a crowded time there. Crowded roads, so being in there decreases your time spent driving each day. We stayed in the park at Madison campground. We loved it! Sites were spacious and natural, very scenic. The park is so big but found this to be somewhat central and allowed us to easily explore Grand Tetons also. No full hook- ups there but that was okay with us. We did take a look at Fishing Bridge campground while we were there - it's the only campground in the park with full hook- ups. Even though I had read up about it, I was shocked. You're as close to your neighbor as you are in a Walmart parking lot with your car. Not the yellowstone experience we were looking for.
As for other recommendations, we took a hike ( a walk really, it's totally flat) on Lone Star trail. It's a 5 mile path ( roundtrip) to lone star geyser which erupts every 3 hours. Impressive geyser. Much less touristy than Old Faithful.
Bring a lunch and relax.
Enjoy your trip! It's a beautiful area.