If you want to travel from Taipei Main Station to Taoyuan Airport, your best option is by far taking the MRT (metro).
The MRT ride from Taipei Main Station to Taoyuan International Airport only takes 35 minutes. This is much faster than taking the bus or a taxi. The bus might be cheaper, but you will probably get stuck in traffic, especially if you travel during the week.
I only recommend taking the taxi if you need the convenience of door-to-door transport, or if you are carrying heavy luggage.
How much does it cost to travel from Taipei Main Station to Taoyuan Airport by MRT?
The fare for a single journey trip from the Main Station to the airport by metro is 160NT$.