Two things everyone should do if they visit Salt Lake City: Since it is "Mormon" country, get to know the culture with a free tour of Temple Square. You will marvel at enormous, gothic structures and learn a little about the predominant culture. The guides are friendly, knowledgeable, and more than willing to share their views and thoughts with you.
Another highlight is the Great Salt lake itself. Next to the Dead Sea, this is the saltiest body of water on Earth! The lake's shoreline fluctuates, but there is always "beach" access somewhere, usually around "Saltair," a local "resort." Take a swim and TRY to sink. It is quite a thrill...trust me!
Quick Tips:
As with anywhere, get to know the culture. Get off of the "main drag" and explore. "State Street" has numerous antique and pawn shops, "The Gateway" is a shoppers paradise, and if you can get to "Jordan Commons" just 70 blocks south of downtown, or so, the ambiance of the Mayan Restaurant (and others) is worth the trip. The Arboretum at the University of Utah is fun anytime of the year, and Hansen Planetarium (downtown) offers several "Laser Shows" each day. And this is just the city! Remember that you are surrounded by mountains, and hiking/camping is available literally on the front door step!
Best Way To Get Around:
Although downtown Salt Lake is criss-crossed with easy and cheap public transit, once you venture outside the city, public transit becomes less frequent. There is a "Free Fare Zone" around the center of the city and the "main" city attractions.
Unless you plan to stay ONLY in the city, a rental car is a must. Taxis are available, but a car is a necessity in the Western U.S. With a population of barely 3 million people, traffic is minimal, even during the "rush hour."