Our family is headed west in June. We are going from PA to the southwest. Driving via I-70 to Utah, to Arches, Canyonlands, 4 Corners, Monument Valley, then Grand Canyon, Las Vegas, Zion, Bryce, Salt...
Make campground reservations at Grand Canyon, Zion and Yellowstone National Parks now. You shouldn't have any problems driving in the mountains. And I totally agree with skipping Four Corners.