Harry Reid International Airport, Las Vegas: The sensible transfer is a rental car, not a one-way taxi or an improvised bus chain. Expect roughly 3¾–4½ hours to Orderville, depending on the route and stops; rental pricing is highly seasonal, but budget roughly US$60–150+ per day once taxes, insurance choices and fuel are in the picture. There is no useful direct public-transport link to town.
Salt Lake City International Airport: Rent a car at the airport and drive about 4½–5 hours south. UTA rail is useful inside Salt Lake City (US$2.50 one way per the airport), but it does not solve the final 300-mile problem: Orderville has no rail or practical airport shuttle connection. SLC is worth choosing only if its flights are much better or Utah is part of a longer road trip.
St. George Regional Airport: A rental car is again essential. It is roughly 1½–2 hours to Orderville and is the most convenient southern-Utah flight gateway, though usually with fewer flights and higher fares than Las Vegas. Do not count on taxis, rideshare, or public transit for this trip.
Cedar City Regional Airport: If the limited flight schedule works, rent a car and allow about 1¼–1½ hours. It is closer than Las Vegas but is not a transport hub; there is no train station or reliable onward public-transport option for Orderville.