If you find yourself looking for more to do when in Park City after the ski areas have closed (keep in mind many years you can find skiing at Snowbird into the summer), you may want to take some short side trips.
Not far and really cool. This includes a hike up the side of a mountain on a twisting paved path (not for the elderly or anybody pushing a stroller). Watch your head. This is a good half day activity when staying at #AbbottSkiHouse in Park City.
Take a longer trip to Moab and tell us what you found. This is an exciting location for all ages. That link will give you a list of happenings. Visit the parks, Moab is a great headquarters for Canyonlands and Arches National Parks. Don't forget to check out the different types of river tours like jet boats and rafting. There are plenty of day trips available.
Goblins Valley State Park
This is a quick hit park you can do while eating a sandwich. You can Clark Griswald it from the parking lot, or get out and walk around. How many movies have you seen this place in?
I've done these slot canyons that are very near Goblins. Slots are kind of cool, you can make your stop as short or long as you want to, but don't go in the slots when rain is in the forecast. Trust me ... flash floods are crazy in these things.
They are hidden all over southern Utah and purposefully kept a secret. They are not advertised so explore some blogs and befriend a park ranger, it will be worth your effort.
Half in Wyoming and half in Utah, not unreachable for a day, but I would suggest a stay over. Bear Lake is good year round, but @JabbottParkCity likes to head to the north end of the lake and drive out onto the sand and make a day of beaching. Swimming and sunbathing in the endless sand are great for all ages.
Mirror Lake Highway
Short drive from Park City. This will take you up and over the Uinta Mountains. It is a nice drive where you can take some hikes, paddle a canoe, go off-roading and do some rental car abuse, check out some little lakes, catch some fish .... you get the idea. Grab a bite at the Smoke House on the way up. I like the pull pork.
When in Park City, you aren't that far from the Teton's and Jackson Wyoming. If you decide to head up there don't miss The Blue Lion. Get the lamb... trust me.
Finally, the National Parks. You can research all of these on your own, but I will give you my two cents.
Capital Reef is not a bad hit on the way through. You can see some rock drawings firght off the main road near the headquarters. Hikes are everywhere. This may not be your first destination, but at least do a drive by, you may come back later.
San Ragael Swell is not a park, but it is about the remote place in the middle of our country. Check a cell phone map if you don't believe me. There are ruins, cave drawings, uranium mines, wild horses and more. All of these attractions are hidden in plain site by a giant expanse. This one is for the adventurous.
Arches "liked it so much, we put it on the license plate". Take a bunch of short hikes and see some arches. Bring a camera and cross it off in one day.
Canyon Lands is HUGE. Don't miss the spires. One of the coolest parks not that many people know about. Probably best seen from a float trip if you have the time. Otherwise drive and hike a bunch.
Zions a little cliche but I like the Angels Landing hike. Hang on, people fall to their death fairly often here. Hit this on the way to Vegas?
Bryce Canyon is fast, easy and panoramic. You don't need to hike to see this place, but I recommend you at least do it once. All of the area around this park is great. Camp, four wheel, hike, fish, hunt, my favorite part of the state in some ways.