Definitely take a walk around Park City's Main Street, home of the Sundance Film Festival. It has lots of restaurants, art galleries, shopping, and bars.
Activities include: scenic gondola rides, hikes, alpine slide, mountain coaster, 400+ miles of mountain bike trails.
Check out this map of restaurants and shops in Canyons Village, where many of the hotels are located.
The Park Silly Sunday Market is an eco friendly, open-air market and street festival. It features local and regional arts and crafts, music and performance art, antiques, imports and one-of-a-kind finds, delicious gourmet foods, and farmers market fruits and vegetables.
The Utah Olympic Park is a sports park built for the 2002 Winter Olympics. They now have tons of family-friendly summer activities.
Wasatch Adventure Guides can help book outdoor activities including: private hiking tours, mountain biking lessons and tours, private water fly fishing trips, horseback riding, skeet shooting.
Located near the Pendry Hotel, Compass Sports can help you book outdoor recreational activities, including guided hikes, mountain biking, alpine slides/coasters, wildlife photography tours, etc.
Park City has several hiking trails of varying difficulty so there's something for everyone! The link will take you to the "Top 13 hikes in Park City"