Salt Lake City (SLC) International Airport is the closest major airport and is about an hour drive to Provo. A car is the easiest, fastest route to the Temple. Ground Transportation options are available from the airport. Once you reach Provo, you would likely need to schedule a taxi/shuttle as they're not readily available in this college city.
Public transportation using a combination of TRAX, Frontrunner and bus is an option to reach Provo. You'd take the Green TRAX line from the airport. Switch to the FrontRunner train at any of the transfer points along the route. Take FrontRunner to the last stop at Provo Central station. A transfer to Bus would be required from here.
Provo Municipal Airport located at 3421 Mike Jense Pkwy, Provo, UT is less than 20 mins. away from the Temple and could be an option if you're flying with Allegiant Airlines.