Late May is still a bit early for some of the higher mountain trails but you should still find good options. You can go to www.MountainTrails.org for information.
My primary recommendation would be Rob's Trail which is one of the most popular in the community. Access is just off SR224 heading into town. You can find it on the trail map. It winds for a mile or two through forest then opens up onto the slopes at Canyons. You can go as far as you like then turn back or use an alternate trail. You should be fine on trail conditions with no snow and not likely any mud.
Another great hike that will give you huge scenic vista is Empire Pass to Jupiter Peak and back. It's a full ridgeline hike atop the Deer Valley Resort that winds up climbing Jupiter Peak. All that said, this route is still likely to have a fair amount of snow and not sure the road will be open. But worth checking.
The Mid Mountain Trail brian_c_pg noted is another good one. A lot of it is shaded and at higher elevation (follows the 8,500 foot contour) so follow the condition on mountaintrails.org before you head up there. The best way to do the trail would be to take the free city bus up to the Montage/Empire Lodge stop then hiking back towards Park City Mountain resort and down into town. I'm thinking late May should be free of snow in most places but likely a bit of mud still. This is a relatively flat trail - some up and down but not a ton of vertical as its objective is to hug the contour.
Also watch to see if the Guardsman Road is open from Salt Lake City. This would be a bit early as they are slow to open it. But if it is open, best option is to take Big Cottonwood Canyon up to Guardsman Pass and down through Empire Pass into Park City. That opens up a world of hiking options for you in Big Cottonwood Canyon. Here's link to closures Look for the SR224 update here. This forum might be better for latest updates.
udottraffic.utah.gov/CLALertViewer.aspx…
Have fun.