Hi Tim, and welcome to Utah! Utah tends to be a very car-centric place, but in the SLC metro area, public transportation is decent. As far as riding all winter, that can be a challenge at times. Fat bikes are becoming very popular for winter riding, but I don't recall seeing anyone commuting on one. It probably happens, however. We do have the "Greatest Snow On Earth", and sometimes lots of it, but for the most part you may be able to get around by bike. We are hoping for a lot of snowfall this winter because we are desperate need water because of very dry conditions the past two years.
The greater SLC area is a mountain valley, but hills are something you will likely have to deal with. What part of the Salt Lake are you moving to? Working at?