Lots of options and various intensity levels around Park City - our favorites are Armstrong Trail (trailhead at Silver Star Cafe), and Blood's Lake at Guardsman's Pass. We recommend downloading the free AllTrails app to search for hikes/trailhead locations based on your skill level! Also - bring water and a layer, it can get chilly the higher in elevation you go!
Rent a mountain bike or electric road bike at Cole Sport right next to Main Street - reserve in advance, and take the many bike paths around to explore PC. If you'd like to do some mountain biking, check out paths at Round Valley Park or up Armstrong Trail. You won't get lost - everything in PC is a loop.
Go check out the history of Utah's 2002 Winter Olympics - probably one of the coolest (no pun intended) museums with history of ski racers, freestylers, lugers, and ice skaters.