Taxi is easily accessible around the different cities in Taiwan. Like most country, the price fare is charge basing on the Taxi Meter once you hop into the taxi. Different City in Taiwan are charge differently but the difference won’t be huge. For long distance journey that are not within the city, it will be good to discuss with the Taxi driver on the charges for the trip and their availability.
Here’s an estimation on how Taxi Fare are being Calculated in Taipei?

First 1.25km will be NT$70
Subsequent 200m will be NT$5
After 12am, an additional of NT$20 is chargeable 

Long Distance Journey not within the City

It will be good to make plan and discuss with the driver on their availability and the charges for the trip.

Airport Pickup/Send Services

A one-way taxi fare between Taoyuan airport and Taipei will cost between NT$800 – NT$1200 from the airport. Airport taxis charge according to the meter in addition to a 50% surcharge (highway tolls not included).

Taxi Booking Services

Taiwan Taxi (台灣大車隊)
Booking Hotline: 405-88888 or 55688 (Press 9) or +886 240588888 (For oversea number)
Customer Service: 0809090993
Official Website

KS Taxi (中華衛星大車隊)
Booking Hotline: 07-7258777
Customer Service: 07-7131000
Official Website