You state you want to park your RV and just travel by foot. Does that mean you will not have another vehicle to drive to different areas? Do you just want to walk from your campsite to see Yellowstone?
If so, you will not accomplish seeing Yellowstone. Yellowstone - and the Tetons - are huge. There's no way to see the important areas from your campsite. You are going to have to drive to certain areas.
Explore this Yellowstone site for 'Things To Do' and you will see that you're going to have to drive to do those things. The Tetons and every other national park has these wonderful sites with lots of information.
http://www.nps.gov/yell/index.htm