Generally our roads are plowed and sanded quickly after a storm (or treated with mag chloride before the snow falls, depending on the temperature). I'd recommend carrying chains and avoiding driving while it is snowing, without proper snow tires. Better to wait until the roads are cleared than to press on and get into trouble. Check out cotrip.org for road information and web cameras all over the mountains.
As spinbackwards said, here in Colorado it is expected that you will have appropriate equipment for the conditions. There aren't checkpoints but if you get into a weather-related wreck without proper equipment you will be cited. I presume that it is much the same in Utah, but don't know for sure.
In general, RWD plus snow tires works quite well — that's what I am using and my steep driveway and road have been snowpacked for weeks. RWD plus chains also works but is a nuisance. However, chains would be more practical for infrequent visits to the mountains than snow tires in Texas.
The cautious driving suggestions by others above are very important when driving in slick conditions. Be aware that bridges can be icy even when the surrounding roads aren't. Also, remember that shady spots can also be icy even when the rest of the road is clear. Overall, just slow down and keep a big following distance when conditions are difficult. If others want to go faster, just let them pass.
