Is this a trick question?
Your route depends on if you want to get there directly via interstates or do you want to get off the interstates to do some siteseeing?
One interstate route based on leaving from Shreveport is I-20 west to I-35/135 north at Dallas to U.S. 81 in Kansas to I-80 west at Lincoln, Nebraska to Salt Lake City, UT and then I-84 to Boise. That is 1895 miles.
Another is from Dallas and I-35/135 to connect to I-70 west to Denver and I-25 north to Cheyenne to I-80 to Salt Lake City, UT to I-84 to Boise. That is 1880 miles but the Denver area is awful.
You could also head to New Mexico and take in some nice sites and southwestern Colorado - Durango/Mesa Verde to the southern Utah national parks (Arches, Capitol Reef, Bryce). You'd be taking some state highways which are good for RVs.
Have a great trip!