I have spent almost 70 years in the "west". Twenty years in Washington State. We're finally going to visit the San Juan Islands this fall.
WITHOUT A DOUBT, the best time to go is after Labor Day. First, decide on whether your focus is wildlife, natural history or history! You could EASILY spend 6 months doing any of those (or others). Yellowstone alone could be a week in October. We feel Glacier is a short car visit or a walking/hiking visit. Lots of people in some areas. The tour on Going-to-the-Sun is well worth it=I'd recommend the shorter tour.
For Yellowstone, picture a big figure "8"==figure at least one day for each loop. Don't forget Cody and the Buffalo Bill Museum=east of Yellowstone=Beware pass may be closed! Beautiful scenery along the Chief Joseph Trail north of Yellowstone. Grand Teton is a nice drive but overshadowed by it's big brother. If you can, stay at the Old Faithful Lodge=the largest log building in the world! Great food!
Devil's Tower and Mt Rushmore were VERY disappointing. Better views from the highway than inside the park/monument.
Washington: Mt Rainier, North Cascades (If the leaves are turning, do the Cascade Loop through North Cascades Highway and back through Stevens Pass or Mt Rainier (if passes aren't snowed up). Quaint town: Leavenworth.
Portland and Washington: Columbia Gorge & Multnomah Falls.
Eastern Washington: The Channelled Scablands and Missoula Flood area. Slightly north is the area depicting the southernmost edge of the glaciers during the Ice Age (one of them). Those large rocks in the fields are "erratics" left by either the Missoula Flood or the glaciers.
As an ex-patriot, "STAY OUT OF CALI!". Driving, timing, etc. makes it an experience to be avoided unless you know what you are doing.
Arizona: Grand Canyon and "Indian Country". If you have time, visit the north rim. Lots of Anasazi ruins (Navajo Nat Monument, Canyon de Chelly, Walnut Canyon (edge of Flagstaff). Sonoran Desert Museum near TUCSON (correction) and San Javier del Bac (most beautiful Spanish Mission ever). Four corners.
Nevada: Not much! Great Basin and portions of Death Valley. Vegas or Reno are a must visit. Virgin River and Mesquite are nice.
Colorado: Four Corners, Rocky Mountain National Park, Stanley Hotel in Estes Park (Haunted), Pike's Peak,
I would make a list of all the suggestions that interest you, read up on them and then rank them.
Have fun! My suggestions were spur of the moment and I likely left out hundreds more!
TAKE CHAINS AND A HEAVY COAT, AS WELL AS SHORTS AND SUNSCREEN!