I think there is little risk unless you are planning on going into the mountains. Public transit does serve many of the ski resorts, and for those in Big or Little Cottonwood Canyon (i.e., Snowbird, Alta, Solitude, and Alta) there is a slight risk of the canyon roads being closed for a few hours during/after a major storm due to avalanche control needs.

If you are going into the mountains with a rental car, don't do it unless you have chains available and know how to put them on as the highway department sometimes mandates them with no advance notice. But in the city itself, you should be fine as long as you don't drive right during a major storm.

That said, people who have no experience with winter weather driving (which I presume you might not with your Hawaiian moniker) should probably just stick to using public transit.