Was in TPE last summer and talk about heat and humidity! As Violist wrote, there is a good bus service between the airport and the city. It's easy to find and you must buy your tickets at counter before hand. Service is regular. If you are in TPE for two days I strongly recommend you visit the Chiang Kai Shek memorial which you can get to by subway. The memorial is on a huge tract of land and a great example of Chinese architecture and space planning. I found the memorial very serene and tranquil. Another option is the Sun Yat Sen (founder of modern China) memorial which you can get to by bus. This is also impressive from a historical perspective. Both would take at least half a day to visit though I spent the entire day at the CKS memorial returning in the evening for a performance at the symphony hall. Your hotel concierge will be able to tell you which bus/subway to take.

The National Museum is also a good bet if that interests you. It bills itself as having the largest collection of Chinese art and artifacts in the world, so much so the mainland would like it back and they could change the main exhibits every week and still not be able to show everything in a year (or something to that effect). Well I was through there in 3 hours beating my previous record of doing the Louvre in 4 hours! Have a great time!