Temple Square is the big Mormon site—free tours, pretty gardens, and you should see it even if religion's not your thing. It's central to understanding the city.
Hit the breweries because Utah changed those old liquor laws. Proper, Squatters, Red Rock—solid spots. The beer's actually good now that they can make it normal strength.
Hike. Seriously, just drive to Big Cottonwood or Little Cottonwood Canyon and pick a trail. Doughnut Falls is easy and crowded. Lake Blanche kicks your butt but pays off. In winter, skip the hike and ski—Alta, Snowbird, Brighton, Solitude, Park City are all right there.
The food scene surprised everyone. Valter's Osteria, Table X, HSL—these aren't just "good for Salt Lake" places. They're just good. Farmers' markets run all summer if you're into that.
Antelope Island sits in the Great Salt Lake, about 45 minutes north. Bison roam around, the lake smells weird, but the views are unreal.
Winter (December to March), if you ski, powder's famous, and crowds are lighter than in Colorado. Spring and fall (April to May, September to October) are best for hiking, decent weather, and fewer tourists. Summer gets hot, but the mountains stay cool. You'll find the cheapest flights to Salt Lake City in late fall and early spring, outside ski season.
Salt Lake City International Airport just got rebuilt and sits about 10 minutes from downtown. The TRAX light rail runs to the airport for a few bucks. Rideshares cost $15-25, depending on where you're headed. For cheap flights to Salt Lake City, check out a fascinating number of prominent carriers and budget carriers, too. Salt Lake City is not what you think it is. You need to go and see for yourself why it is an incredible destination to explore.