Hill AFB Museum in Ogden is very worthy, if you like that sort of thing. Sundance is nice for a Sunday brunch, or even just a short walk around in later May. There are a lot of wildlife and waterfowl refuges if you are into birding. A romantic dinner at Loghaven up Millcreek Canyon (Make sure they seat you in the main dining room at a table with chairs). Skiing, if you like (x-c or downhill, depending if you are in early May or late May and the snowpack). Walk Park City Main street, especially the upper, historic, part. Temple Square, if you are into that sort of thing. Jazz game, if you like basketball, and lots of great dining and after game drinks downtown. Depending again on the time of month and the snow year, The Mirror Lake Scenic By-Way may be worth a visit, and also the Alpine Loop Scenic By-Way. Timpanogas Cave, if you hike. Let me know if you want details on any of these options.
There is a new Natural History Museum that I didn't get to before we moved away... On the UofU campus. Also Ft. Douglas on campus has some historic buildings and a small museum.
To be honest, I lived there 25 years and spent nearly all of my time trying to figure out how to get out of there... for me SLC was great because it is in the middle of the west, near The Rockies, Colorado Plateau and some Great Basin attractions. In May I would strongly recommend considering S. Utah and the Colorado Plateau if you can.