I've winter camped twice...the best part about both experiences was what happened on the hikes out in the middle of the night after I gave up on the camping!
That having been said, the experiences on the hikes out in the middle of the night were some of the coolest experiences I've ever had in the backcountry...so while the camping itself was pretty miserable, it's hard to completely bad-mouth winter backpacking.
For something close to Salt Lake, really anything works. My experience was that it was hard to snowshoe very far, so you don't need to pick anywhere big. A good place might be the north end of the Mirror Lake Highway...maybe just head out on the North Slope Road and find a place to head out into the woods. Or maybe Nobletts Creek or Log Hollow on Highway 35 outside of Kamas. You wouldn't have snowmobile traffic on those trails, but they wouldn't be packed as much (or at all), either.
As you mentioned, I wouldn't plan on going terribly far your first time, because it may not last all night, and you don't want to have to hike for hours in the middle of the night. Not only that, in the middle of the winter, you will have exactly the same isolation, snow, and temperature if you're one mile from the road as you will 5 miles from the road.