You've received a bunch of good advice already and I would concur with most all of it. We were in Yellowstone last summer after not being there for over 40 years (once in 1968 as a teenager and 1977 as a young adult). Only thing I want to emphasize is to be out early if you want to see more wildlife and have less traffic. The crowds were heavy and parking was limited by 11am or so but that may be different this year with the pandemic because many of the visitors were Asian travelers. We split a week up between Grand Tetons NP and Yellowstone NP. It was good to be back at Yellowstone and we enjoyed it but thoroughly loved spending several days in the Grand Tetons (which I just drove through before) and I would go back again today to Grand Tetons over Yellowstone, but you have to see Yellowstone at least once. In Yellowstone, you can't beat Grand Prismatic Spring, the colors are unbelievable. It's a must see as much as Old Faithful.
As one who grew up in Oregon (all my family still lives there and I go back 1-2x per year) and went to college and grad school for 7 years in VA, lived in upstate NY for 7 years and now in FL (the last 34 years), my wife and I have driven the entire USA from West to East and East to West (north, south and central routes) multiple times. We've been to all 50 states and most of the National Parks (NP). So let me suggest a few other places in your route between AZ and Yellowstone that we love the most: Sedona, AZ, Grand Canyon, Antelope (Slot) Canyons at Page, AZ, along with Horseshoe Bend. Also, Bryce NP, Zion, NP in Utah, Arches NP in Moab, UT would be highly recommended along that route. After Grand Tetons and Yellowstone, I would recommend Glacier NP in MT if you have the time and want to go that far north and west. And then others would Badlands NP in South Dakota on your way back toward SC. Of course, if you have time coming or going, Colorado has tons of awesome places to see in the Rocky Mountains. My highlights over the years have been Rocky Mountain NP, driving to the top of Mt. Evans, and Pikes Peak, and Garden of the Gods.
And then for another trip, unless you have the time this time around, would be to go on out to the Pacific Northwest. I may be a tiny bit biased but the beauty of Northern California, Oregon and Washington is hard to surpass (remember, I've been in all 50 states). The variety of the Redwoods and Sequoias, the Oregon Coast, Crater Lake NP, Olympic NP, the Cascade mountains (Mt. Hood, Mt. St. Helens, Mt. Ranier, etc) are just incredible. I have some images from most of these places on my website at
www.GingerichPhotoArt.com if you care to browse.
Enough for now! Hope that helps. Enjoy your retirement. Congratulations.