Leaving mid-April go to Arizona first and work your way north toward Yellowstone and you'll be just fine for traveling.
For the Grand Canyon plan on staying in the park itself at Trailer Village for the best experience. The free shuttle bus picks you up at the campground and you can ride all day long getting on and off whenever.
In Arizona you might consider Canyon de Chelly Nat'l Park and Monument Valley Tribal Park.
Then head into Utah for these national parks. May is a great time. Zion, Bryce, Capitol Reef, Canyonlands, Arches. Then head further north to Grand Teton Nat'l Park and Yellowstone. Then toward South Dakota.
Three months is a perfect timeframe. Have a good one!