If I were you, I would take ibbarry’s suggestion, take any of the buses that run from Chiang Kai Shek (CKS) to the city of Taipei. CKS is mostly for international flights. Taxi is a lot more expensive than these buses. There are also other buses that take you from CKS airport to various points in the city of Taipei. Most stops are major hotels. The next morning, you can then take a taxi from your hotel to Sung Shan Airport in Taipei. Sung Shan Airport is for domestic flights only.