Don't forget that the HSR station in Tainan isnt at city center (as opposed to the Tainan train station)

If you're going to city center, then you'd have to take a commuter train from the HSR station to the city station.

So your journey will look like this
MRT from Taoyuan Airport to Taoyuan HSR Station
HSR from Taoyuan HSR station to Tainan HSR station
commuter rail from Tainan HSR to city rail station