Taiwan is a beautiful tropical country surrounded by the Taiwan Straight, the South China Sea, the East China Sea, and the Pacific Ocean. Taiwan is a fairly small country at just over 13,000 square miles (comparison 272.9 times smaller than the United States / If Taiwan were duplicated, it could fit into the United States 272.9 times) so it is fairly easy to get to and around using airlines, railways, the subway systems, intercity buses, taxis, cars, scooters, and boats.
There are 3 international airports in Taiwan and 15 domestic airports (total of 12 airports). A full list of international airlines can be found here. Domestic airlines include Far Eastern Air Transport, Mandarin Airlines, TransAsia Airways, and UNI Airways. (*Please note TransAsia Airways and Mandarin Airlines have websites with English options).
Tickets for numerous cities and towns across Taiwan can be purchased online (and in English). For the High Speed Railway, visit this site (serving Taipei, Banqiao, Taoyuan, Hsinchu, Taichung, Chiayi, Tainan, and Zuoying). For other destinations in Taiwan, visit the the Railway site. I personally advise purchasing tickets online before your arrival to Taiwan as seats can sell out quickly during peak travel seasons. It is better to be safe by purchasing ahead of time than to have to cancel your travel plans to a city.
Highway bus service is provided by private companies and a complete listing can be found here.