Generally speaking, the mountains are covered with snow until around May. These are the ones close to Salt Lake. So, if you want to go snow shoeing, I can recommend a few places. Basically, park at the ski resorts and hike on out.

Other than that, there is Antelope Island and Stansbury Island. The best hiking on Stansbury is during the early spring when the mountains have snow and the flies aren't out to make you go insane. It is a treeless, rocky hike with some great views no water and no flies. After June, Stansbury Island is horrible.

Here is a hiking forum that is specific to Utah. Bogley