Watch the weather, as a PP said. As for route, I would recommend I90 to MT 55 to MT 41 toward Dillon, MT. Pick up I-15 there and take it though SLC to St. George. It is a good compromise between taking interstate all the way since you go so far out of your way to hit 15 from 90. However if the weather is bad and you still have to go, stay on the interstate the entire way. It will take a lot longer though. As for going through SLC, as long as you drive it between 9 and 3 you will be fine. If you hit it at rush hour, you will regret it. I live here and I know! I would recommend taking I-15 directly rather than one of the I-215 belt loops.
It is about 5 hours from SLC to St. George. If you decide to stay in SLC, I recommend the KOA on South Temple St. it is right across the street from the light rail that will take you straight to the downtown area with Temple Square and other neat historic sites. Pony Express is also a good one, but no light rail. I don't have recommendations for campgrounds north of there since we generally drive from SLC to Bozeman.