Taipei's C.K.S International Airport is well-served by international flights. All Asian countries have flights to Taipei, with many inter-continental flights touching down in the capital as well. It's worth remembering that many international airlines change their names in Taipei so as not to 'offend' the Chinese, for example British Airways becomes British Asia Airways and Japan Airlines becomes Japan Asia Airlines.
Taipei has an extensive city and regional bus service, based around four major bus terminals (the city government is trying to get them all into one terminal, which makes it a little confusing). If you're travelling to other parts of Taiwan, the service is stunning; 111 buses make the two and a half hour trip to Taichung each day, for example. There are even night buses that are great for going to Kenting or Taidong. You can sleep and arrive in the south in the early morning. The train network is also comprehensive, though a little more expensive.
With Taipei's airport on the northeastern fringe of the city area, getting to town is easy and relatively cheap. Taxis are available at the airport, though the drivers rarely speak English - it's a good idea to have your destination written in Chinese on cards. Driving in Taipei is hectic at the best of times. Unless you've been there six months, it's best to let someone else do the driving.
The MRT (Metropolitan Rapid Transit) system was Taiwan's first urban mass-transit system. The half dozen lines work well, the system is seldom overcrowded and it's inexpensive. Trains run every four to seven minutes from about to .