You can entertain yourself and your kids for free if you're willing to take advantage of free admission days at a variety of Wasatch Front attractions. Utah's Hogle Zoo, for example, offers free admission the last Wednesday of November, December, January and February. The Utah Museum of Natural History offers free admission the first Monday of every month. The Utah Museum of Fine Arts has free admission the first Wednesday of each month. Most organizations offering free admission do so due to funding from Salt Lake County’s Zoo, Arts and Parks program. Red Butte Garden at the University of Utah offers several free-admission days per year. The next is a Holiday Open House & Art Fair, December 6 and 7. Information can be found here. Occasionally shows and other entertainment are offered free and are listed at http://www.slcozap.org/zapArts/html/generalinfo/freeEvents.html.
You can search for free and low-cost family-oriented entertainment at http://gocitykids.parentsconnect.com. Select the Salt Lake City site.