Taipei city is about 20-30 minutes away by bus from the airport, there are no MRT (subway) lines connect it yet. One suggestion is to stay at the Taipei Grand Hotel, that'll put in you ShiLin district, and you can experience of the night markets there in Shilin, great place to see Taiwanese food/night culture, if you like "street food". THere's lots of great stuff to eat, and it's an interesting experience.

There are buses that run from the airport to most of the major hotels. Once you pass customs, you'll exit into the waiting area. Turn left and at the end of the hall will be a number of kiosks for buses. These buses can run both ways, bringing you into your hotel and back to the airport.

If you use a cab, I think it's about $1000 NTD, about $30USD, each way.

If you want to see more modern Taipei (like the Taipei 101 building, tallest in the world at the moment), you'll want to find a hotel in East Taipei, which is much like any modern american city, with bars, restaurants, shopping, etc.

http://wikitravel.org/en/Taipei