The closest international airport is in Kaohsiung. From there you can take a train, bus, taxi, or rental car for a 45 minute to one hour journey to Tainan.
There is a domestic airport in Tainan, and the flights from Taipei take about 45 minutes, although the route is now defunct.
By bus
By bus
Tainan has good inter-city bus connections with other cities in Taiwan.