After a long flight, taking the Taoyuan Airport taxi service to Taipei city might be what you need. The service is fast (except during rush hour), comfortable, and reliable.
Here is a picture of fares from Taiwan Taoyuan International Airport to Taipei and other cities:

You can catch a taxi at Taoyuan International Airport at both terminal 1 and terminal 2. There are two options: you can either go to a taxi counter inside the airport, or catch it outside on the sidewalk at designated areas. This is what the taxi counters look like:

The great news about the Taoyuan International Airport taxi service is that staff can speak English and they will be able to direct you to the right place to catch your taxi. You can tell them your hotel name in English and they will do the translation work for you to the taxi driver.
Another awesome news is that you can use a credit card for payment for your taxi ride, so you don’t have to change your money at the airport if you’re in a rush.
Taiwan Taoyuan Airport Taxi Fare
Estimated fares from Taoyuan Airport, in the local Taiwanese currency (NT$).
- Taipei City: 1100 – 1500$
- New Taipei City: 900 – 2000$
- Keelung City: 1500 – 1800$
- HSR (High Speed Rail) Station: 330$
- Taoyuan City: 500 – 1200$
- Hsinchu County: 1200 – 2000