There are a ton of great options in and near Park City.
Here are some possibilities
1 day in Park City - Mid-mountain trail.
1 day Guardsman pass to Big Cottonwood Canyon -hike Silver Lake
1 day Mirror Highway. Walk around Mirror Lake - 2 miles. Check out Provo Falls.
1 day Alpine Loop - Provo to American Fork - hike Scout Falls from Timpanooke campground. Check out Bridal Veil Falls and Cascade Springs. Maybe hike and tour the Timp Cave.
None of these have exceptional odds of wildlife other than deer - you may see a moose on the Scout Falls hike - my son saw a couple yesterday. You could drive to Antelope Island for bison.
There are more than a few good trails at the Snowbird Resort accessed from Salt Lake. You can take the Snowbird Tram to the top of Hidden Peak and hike down the tram ride is $20. I hiked the Peruvian Gulch trail down a few weeks ago and it was beautiful, a 4 mile downhill calf crusher. Hike it on a weekend and you can catch Oktoberfest at Snowbird.
If you do the Alpine Loop as suggested above, Timpanogos Cave is a must. It's 1 1/2 miles up to the cave on a paved trail but it gains 1000 feet in elevation so it get's a little steep in spots. Once you get to the top of the trail you get a cave tour from the Park Service. This will cost $8 each, well worth it, the views from the trail are spectacular and the cave itself is beautiful. Another good waterfall hike on the Alpine Loop is Stewart Cascades aka Stewart Falls, it's a two miler each way and goes up and downhill coming and going; it's not far from the Sundance Resort.
If you're in town before the 17th make sure you're in Park City for the "Silly Market" at the bottom of Main Street on Sundays; one of the town's most popular summertime events.
Good advice so far. A few points to add. Hundreds of miles of trails in Park City area. Checkout www.mountaintrails.org. A few of our faves are Rob's Trail - nice easy hike through stand of aspens up into bowls of Canyons resort area. Super favorite is ridgeline hike from Empire Pass above Deer Valley over to Jupiter Peak at Park City Mountain resort.
I would be cautious in switching an entire vacation plan due to smoke. Smoke in any given area is relative more to weather and wind patterns and can change in a flash. We're actually pretty smoke right now in Park City. But it could change in an instant.
For Stewart Falls - there are 2 ways to do the hike. The longer route is about 4.5 miles - starting from the parking lot just after the ranger booth where they collect the $8. The shorter, easier way is to ride the ski lift up at Sundance and then hike to Stewart Falls (I've never done it this way and don't know the mileage.)
Thank you everyone for all your helpful information.
Would you recommend any of these hikes in mid April?
megdohOT, you should start a new thread as really nothing is comparable from this one to mid-April. April is still winter with pretty much everything noted here still snow covered. As noted in your other thread, mid-April is between seasons and you'll find pretty limited access and seasonal closures.