I would take US 89 from Grand Canyon through Page and Kanab up to Bryce. They have very nice campgrounds at Bryce. Upon leaving Bryce, return to US 89 and go north all the way to Nephi where you pick up I-15. Take 15 to Idaho Falls where you will pick up US 20 into West Yellowstone. There are many nice campgrounds in West Yellowstone. Leave the coach there and spend 2-3 days exploring the park. When you leave, cut across the center of the park and exit through the east entrance over to Cody and then work your way across to Gillette. It is about 575 miles from Bryce to West Yellowstone. There are some very nice RV parks in the Salt Lake area. It is about 325 miles from there to West Yellowstone. The Grizzly RV park in West is excellent as is the Ponderosa in Cody. You may want to consider spending a couple of days in Cody checking out the Buffalo Bill museum. It is incredible.