I travel from Tucson to Salt Lake several times a year to see our son. Our preferred route is to take I-17 to I-40, then take exit 201 to HWY 89 North. You'll go through Page AZ, Kanab UT and several other small towns along the way. The scenery is beautiful. Follow HWY 89 until you get to UT-20. Turn left on UT-20 for approximately 20 miles and then head north to Salt Lake on I-15.
I've driven this route during Christmas for the last 12 years, and we've only encountered icy conditions once during that time. The chance of bad weather closing the route is getting more and more unlikely as we move farther into Spring. The road crews in Utah are quick to plow and salt if needed. However, if a late Winter storm hits, the best route is through Las Vegas.