Travel to Kauai from the "mainland" (the rest of the United States) isn't cheap. Airfare from Salt Lake City to Honolulu ranges from $500 to $1400 round-trip depending on the carrier, time of year, and how early you book your flights in advance. Less expensive flights are available if you are flying from the west coast.
Hawaii is one of the most popular vacation destinations in the world, and unfortunately the airlines choose to keep these flights full rather than add more flights. A large percentage of people traveling to Hawaii are using frequent flyer miles to get there. Don't expect to find a lot of vacant seats on the plane. Most flights are sold out by departure date.
Lihue Airport (LIH) pronounced lee-HU-ay ("ay" as in hay), handles virtually all arrivals on Kauai. The terminal in Lihue is small but quite modern and efficient. Airlines that fly directly to Kauai include Delta, Hawaiian, United, US Airways (from Phoenix), and Alaska Airlines (from Seattle). Because of the limited number of airlines servicing Kauai, many people fly other airlines to Honolulu and then use an interisland carrier to travel to Kauai.
Honolulu International Airport (HNL) is a large airport and is serviced by airlines from all over the world. Inter-island flights to Kauai, Maui, Molokai, and the Big Island are located in a separate smaller terminal on the far western side of the main terminal. To get to the Interisland terminal, turn left at the curb when leaving the baggage claim area and follow the walkway. Rent a "smart-cart" for $1.25 if you are carrying luggage, or take the free "Wiki-Wiki" bus at curbside because it's a fairly long walk, especially if you are coming from United's gate which is at the far opposite side of the airport. Delta's terminal is much closer.
At the Inter-island terminal in Honolulu, both Hawaiian and go! airlines operate flights about every hour to Kauai. Ticket prices vary from about $60 to over $100 each way. If you arrive earlier than your reserved interisland flight, ask to go stand-by on the next available flight to Lihue. You might save yourself 40 minutes of waiting. Keep in mind that if you do this, you will probably arrive before your luggage (unless you checked it only to Honolulu and then re-check it again on your flight to Kauai). If you do arrive before your luggage, just relax and pick up your rental car while you wait. Your luggage will probably be at the baggage claim in Lihue by the time you pick up your car. Just don't make the mistake of packing your rental car information in your luggage. It's better to keep this with you so you can pick up your car even if your luggage is delayed.