The international airport in Tahiti is called Fa'a'ā. It is a 15-minute drive away from the city center but there can be high traffic in the direction Airport->Papeete early in the morning up to 8am, and starting at around 3:30pm until 5pm in the direction Papeete->Airport.
Airport - Papeete
There is a bus stand directly across the street from the airport. Buses run randomly weekdays and Sat during the day.
Papeete - Airport
You can catch the bus near the central market. Mon-Sat during daylight hours
There is a taxi stand right in front of the Papeete marina (the one in the city center) on the other side of the road very close to the Vaima shopping center next to the Rétro bar.
Depending on time of day taxi's from city center run about 20-25$. Bags also contribute to cost. Also the number of people as well will increase the cost