Best cheap thing to do: Hike. You are never more than a half-hour from a mountain. For a short jaunt, there is a canyon that you can explore right Downtown, next to the City Creek Mall.
Why it's worth a trip: Perpetually, Salt Lake City is going to be overlooked by tourists -- it's in a landlocked state, it's notoriously religious -- but that just means less competition for the best finds. Take, for instance, vintage shopping. It's amazing here, and never better than at Decades, one of the best shops in the country for vintage and antique clothing. I picked up a Edwardian wedding dress for a mere $80, a price that ensures whale-bone corsets will make a comeback soon.
If you are planning to stay a few nights, forego the hotels and crash in one of the Downtown Victorian B&Bs -- you can find rooms at the Ellerbeck Mansion, for one, at less than $130 a night. Romantic and quaint, with four poster beds and lots of floral, even a hard-hearted city dweller will be charmed.
Since SLC hosted the Olympics in 2002, getting around has been made incredibly easy. The Trax trolley system runs all over town, with stops at the airport and the University of Utah campus. There are also buses, but Uber here is cheap and plentiful. This can prove useful should your legs feel too wobbly after one of those mountain hikes or if drinking at the high altitude proves challenging, which, spoiler alert, it definitely will.