In three days you'll be hard-pressed to see all of this region so your best bet is to stick to the central Wasatch. Start with a day and a night in Park City, skiing one of its three resorts in winter or taking advantage of lift-served mountain biking in summer, then enjoying a night out on historic Main Street. The next day drive west on I-80, south along Salt Lake City's eastern bench on I-215, and east again on Route 210 up Little Cottonwood Canyon. In winter spend a day skiing at Alta or Snowbird. In summer hike the Catherine Pass trail into neighboring Big Cottonwood Canyon. On your final day head south again on I-15, then east on Route 92 on the Alpine Loop scenic drive to Sundance Resort, where you can pamper yourself with a massage at the Spa and a quiet meal at the Tree Room or the Foundry Grill. In winter the Alpine Loop will be closed to traffic, so instead of driving east on Route 92 continue south on I-15 and spend some time exploring Provo before driving east on U.S. 189 to Sundance.
Start your five-day trip by following the suggested three-day itinerary above and then tack on the following: on Day 4 leave Sundance, heading east on Route 189 up Provo Canyon to the Heber Valley. In winter spend the day cross-country skiing or snowshoeing before your late-afternoon sleigh ride, and in summer go hiking, golfing, or horseback riding in Wasatch Mountain State Park. The point is to take in the beauty of the Wasatch at a slower pace than downhill skiing or snowboarding allows. Treat yourself to supper at Snake Creek Grill before retiring to the overnight comforts of the Homestead Resort. Strike out on Day 5 using River Road to connect to U.S. 40 and back to Park City where, even though you started your trip here, something new awaits. Ski and/or snowboard until you're worn out and then nestle into an après-ski lounge for a hot toddy before suppertime and your pre-sleep hot tub. In warmer months spend your day fly-fishing with one of the guides from Trout Bum 2 or hiking some of Park City's extensive trail system and cap it off with supper on the patio of one of Park City's fine restaurants.
In a week you can get a good feel for the diversity of the Wasatch. Follow the five-day itinerary above, adding a day and a night in Big Cottonwood Canyon between Park City and Little Cottonwood Canyon. Then you can spend an extra day and night either exploring the Heber Valley or Park City.