Don't fly to Tainan Airport - would avoid it anyway possible.
Why? First, the flights are fewer and fewer. So you more than likely have to wait for your flight in Taipie or Kaosiung. There is rarely a savings in money or time in taking a flight.
Both highspeed train and TNN are out of the city so both methods involve taxi costs. TNN is closer to city central, while the HSR is close to the eastern part of the city. That been said the cost of a taxi would still be fairily inexpensive.
The flight itself is bumpy and cramped. You will change planes to a small commuter plane.
Unless you are in for the experience, the HSR is so much better option.
Any why is this in the China section?