For Yellowstone, your best experience will be to stay in the park itself. Madison is a great campground.
You won't be able to stay at Fishing Bridge because hybrids are not allowed - lots of bear activity there.
If you can work in 2-3 nights at Grand Teton Nat'l Park which is connected to Yellowstone to the south, I think you would love it. Excellent hiking for all abilities on beautiful trails and the looks of the park are completely different from Yellowstone - like a mini Alps! You can drive through the south entrance of Yellowstone to enter Grand Teton.
Have a great trip!!
