Salt Lake City loosened up it's liquor laws 10 years ago and the cocktail and alcohol culture has flourished ever since. Which isn't to say that they still don't have the weirdest liquor laws anywhere but at least you don't have to "join a club" just to get a drink these days.
High West Distillery
If you can't make it out to Park City to see the gorgeous distillery... and I seriously recommend that you do... you can visit the High West Saloon at the airport. They had great craft cocktails, spirit forward and generous pours. To to clear, this is most likely the only time you will ever get a recommendation from me to visit an airport bar!
Whiskey Street
This whiskey focused bar features the largest selection of whiskey in the state of Utah with around 350 bottles. Named for the area that once was full of distilleries in the 19th century. oh, and have the pork belly corndogs, just do it.
The Red Door
In a city best known for it's religious affiliation Red Door has taken an irreverent approach to it's decor and it's cocktails. It's situated next to the Monaco Hotel which is my favorite place to stay when I am in town.