Reddit mentions of Wasatch Winter Trails

Quite close to you: Bells Canyon at least, the lower part of the trail near the reservoir. It's rare that there's so much snow in the valley to make that area dangerous. A bit farther away, Bonneville shoreline trail near Red Butte Garden is pretty nice when the LCC/BCC trails are covered in snow.

Otherwise, I'd suggest investing in a pair of used snow shoes or "golf course" cross-country skis. With an investment of <$100, you can ski safely on terrain with basically zero avalanche risk. Some of the better ones are Lambs Canyon, lower Neffs Canyon, and Guardsman Pass. For these options, I recommend the book "Wasatch Winter Trails" by John Veranth.

http://www.amazon.com/Wasatch-Winter-Trails-John-Veranth/dp/0874806291

For any of the incredible BCC/LCC hikes on more difficult terrain, you want to look into snowshoes, telemark or alpine touring gear, an emergency backpack with avalanche gear, and (most importantly) competent friends willing to dig you out in case of avalanche. That requires a significant investment of money, and (unless you already ski) time learning to ski the backcountry. For this, I recommend getting secondhand tele gear (it's so much cheaper than AT, and more fun too), and taking an backcountry/avalanche course in one of the resorts. And buy the seminal Hanscom-Kellner book, "Wasatch Tours":

http://www.amazon.com/Wasatch-Tours-The-Northern-Volume/dp/1884744036