Much of Utah's desert areas are clear or get scattered snow, but are excellent to hike when the sun isn't beating down. Remember to pack water, since the air is just as arid, even when temperatures are lower, and forgetting to stay hydrated is easier.
The Chinle Trail, just outside Zion Park boundaries, follows a dry wash. When the trail splits continue on the road about three miles to hike into pinion forests and a scattering of petrified wood. Continue on to between Scoggins Wash (may have running water) and Coal Pits Wash to Old Scoggins Stock Trail to Coalpits Spring's winter waterfall.
Further afield, Bryce Canyon and the Grand Canyon offer great cross country skiing through areas that are even more picturesque when snow capped. It isn't unusual to see deer track or the deer themselves and well as other wild creatures like rabbits, chipmunks and wintering birds.
Utah is also noted for its significant mountain ranges. The La Sal Mountains, near Moab, are criss-crossed with trails that are suitable for cross country skiing during its elongated snow season, often from early fall to late spring.
When you drive out of Salt Lake City, the nearby Wasatch Range has endless opportunities for outdoor recreation, hiking, and, when it is the season for snow, both downhill and cross-country skiing. The area contains seven large ski resorts, Alta, Snowbird, Solitude, Brighton, Park City, Deer Valley, and the Canyons. The mountains around Park City have some of the deepest snow and best powder in the world. Just a bit farther south is more great skiing at Brian Head near Cedar Breaks National Monument.
Throw in some ice fishing, and you may never come to Utah in summer again. Check out Panguitch Lake, Lost Creek Reservoir, Otter Creek, Current Creek Reservoir, Minersville Reservoir, but Fish Lake may offer the best ice fishing in the state. Navajo Lake is often inaccessible during ice fishing season, but the fishing is sometimes great for those who can get there.
Salt Lake City hosted the 2002 Winter Olympics and you can visit the Olympic Park. Ride the Comet Bobsled, go extreme snow tubing, or watch elite athletes train. By the way, you can thank the Olympics for the city's improved public transportation. Public support rallied for better public transport after the town was designated as the Olympics site and the Utah Transit Authority was finally able to complete a 15-mile light-rail transit line called the TRAX, among other improvements.
If you are headed to the Grand Canyon, drive through Monument Valley on the way. The snow-covered rock formations make the road trip memoriable.