I've done the bus - 3 one-way in a row because my kid left his phone on the inbound EVA flight and we had to bus back to TSA to pick it up. Bus is efficient, at TSA there may or may not be a person there collecting fees, if not you'll put it in the fare box in front of the driver. Either case you'll need NT$, the bus is parked just outside of TSA and is easy to find

http://www.taoyuan-airport.com/english/buses - "1840 to Songshan Airport"