Hotel shuttles and taxis connect the airport with downtown. I made sure our hotel had a free shuttle as I didn't want to pay for a taxi even though the airport is fairly close to the downtown.
If you don't have too much luggage, UTA buses offer an inexpensive $2 ride from the airport to downtown. But free is better.
I flew in and out of Salt Lake City Airport which is just west of downtown and just south of the Great Salt Lake. This is the only major airport in the region and is Delta's second largest hub. In recent years the airport has handled some 22 million passengers, making it America's 22nd largest airport.
Southwest has a great schedule but a no frills airplane. Getting to SLC was tough. I have never experienced a delay or missed a flight ever! So, to have both happen in the same trip was discouraging. Oh well, we endured. SLC airport is pretty organized. Not too big.
When I arrived in Salt Lake, another traveler commented to me that the airport was going to be a giant mega-bottleneck during the Olympics because it was so inefficiently managed. They couldn't even tell us which carousel our luggage would arrive on.
We only got to see the 2002 Winter Olympics on TV, so that didn't worry me too much. Bob was recovering from his heart attack, and it was an interesting thing to watch at that time.
Conde Nast Traveler magazine ranked Salt Lake City International Airport the ninth best in the U.S. The airport's new International Building has customs and a duty free shop. Salt Lake International City Airport is closer to the heart of the city it serves (a ten minute drive) than nearly any other U.S. airport.
Nine major airlines and two regional carriers serve Salt Lake, offering a total of 624 arriving and departing flights daily, with 75,000 passenger seats. As the 'Crossroads of the West' at least half of America's population is located within a 2 1\2 hour flight from Salt Lake.
Salt Lake is of course a Delta hub so that is always an option; however I often find flights cheaper on other airlines. Also they just rebuilt the freeway for the Olympics and it is easy driving.
If you are going to be just downtown or have prearranged transportation, prearrange for a shuttle service to take you to the hotel. It is much cheaper if you are a small party of one or two. Most taxi fares start at a twenty dollar minimum and a shuttle for one is around six to the downtown area.
I highly recommend a car, one with guts to get up mountains, for traveling around Salt Lake and the surrounding mountains. This is another western town in love with the automobile. They did install a light rail train before the Olympics and are continuing to expand its service, but it is still limited in usefulness to the average tourist.