The National Theater in Taipei is the best theater available in Taiwan. It offers a wide variety of programs, including traditional Taiwanese drama/dance and Peking operas, as well as classic operas, ballet, and Broadway-type shows. You can check out the ongoing programs at http://www.ntch.edu.tw/ and purchase tickets online.
The National Theater is located inside the Chiang Kai-shek Memorial Park, one of the landmarks in Taipei. If you have time and the weather permits, after the program, I'd recommend to take a walk in the Memorial Park, where you can enjoy a slower-paced and a more relaxed side of Taipei that you don't find very easily elsewhere in the city.
Quick Note: The National Theater can be easily accessible by TRTS, Taipei's metro system. It is free to enter the Memorial Park, and the tickets to the theater range from $10 to $50 in average. Formal attire is usually not required. Copies of the programs can be found at most of the large bookstores in Taipei. They are also listed online.