Hi, I am located in the SLC area and I am from Denver, we travel back and forth often all different times of the year.
The worst stretch of roads normally is Cheyenne to Laramie, some times around the Rock Springs area. You usually dont see much adverse weather in terms of snow and ice this tiem of year, that area was just hit by a freak storm so it is a little wet/slick but should be clear by Sat.
Here is a good site for road info if you dont have it already. WYDOT Travel Information Service (Laramie)
, I just checked and it is clear/dry now
There are campgrounds or truck stops about every 90 miles in most places. There are only 2 longer hills and then just easier mountains near park city, the hardest obstacle is a steeper downhill in parleys canyon just before you enter SLC