In the winter, if you have only a few days, it’s best to pick an area—Big Cottonwood Canyon [1], Little Cottonwood Canyon [2], or Park City [3]—and base yourself there for skiing and boarding. If not everybody in your group is interested in skiing all day long, Park City is the best bet—it’s a real town with as much activity off the slopes as on them.
In the summer, it’s nice to have a car (or a bike!) and the freedom to poke around the mountains. Increasingly, the ski areas have structured summer activities, but there are also plenty of trails and a real sense of relaxation that sets in. Indeed, the Cottonwood Canyon areas are very quiet in the summer, and they are good places to camp and hike.
Links:
[1] http://www.moon.com/destinations/utah/park-city-and-the-wasatch-range/big-cottonwood-canyon
[2] http://www.moon.com/destinations/utah/park-city-and-the-wasatch-range/little-cottonwood-canyon
[3] http://www.moon.com/destinations/utah/park-city-and-the-wasatch-range/park-city