3-4 days minimum in Yellowstone. The last day be sure to head south through the park to get to Grand Teton Nat'l Park and plan to spend about 3 days there. It's a beauty and altogether different from Yellowstone. In Grand Teton Colter Bay campground has 300 non-reservable dry sites as does Gros Ventre. If you need hookups, Colter Bay RV in the park has F/H/U and is right next door to Colter Bay campground - reservations needed. Staying IN the parks themselves saves you time and gives you a better feeling of the surrounding wilderness rather than staying outside the parks.
For Grand Canyon, definitely stay in the park near the rim's edge at Trailer Village with F/H/U. The free shuttle stops just outside the campground entrance and you can ride it all day getting on and off whenever. It's a great way for touring. Have a fun trip!