It is doable, as noted, but will involve a lot of driving and you will just scratch the surface of all there is to see in the great American West.
Our first trip out there we had one month. We traveled from Virginia to Rapid City (Mt. Rushmore). Then stopped in Rocky Mountain National Park in Estes Park, CO. From there we drove down thru Moab, UT and on into Monument Valley. We then went to the south rim of the Grand Canyon. From there we started back east and stopped in Amarillo to visit Palo Duro Canyon, and then stopped in St. Louis before heading home.
We stayed in each place for several days (2 -4 generally). It was a lot of driving but we enjoyed being on the move and just seeing parts of the country we had never seen before.
Since retirement, we've been back 3 more times but we allow at least 3 months.
I agree with the others about saving Yellowstone for another time. It is just so huge. We spent a week there and still did not see it all.