The most important thing to travel in Taiwan is to know how to speak Mandarin and to read Chinese because most of the people in Taiwan don't know English, but I guess you will still survive if you don't know Mandarin, some of the Taiwan university students do know English anyway. Undeniably, Taiwanese are the most friendly people I have met in my life, you will feel like home.
The most exciting part about traveling in Taiwan is to visit their famous night markets which are located everywhere in Taiwan, mostly near to train stations, the night market food tastes heavenly, you will come back becoming a fatty. Another must-visit-place is their old streets like jiufen, shifen, tam shui and many more, to experience a total different kind of Chinese old and traditional culture. All the places are reachable by taking public transport like train, bus or cab.
This is a shopping heaven, you can shop anything trendy here until you drop.
Its a lovely old street with all the old buildings and fresh air, a total different cultural experience.
Its an old township near ZhangHua, middle of Taiwan. Nothing much here but to experience the old streets, the historical buildings, and the culture here.
Its another old street which is full of delicious food and old shops around, friendly people, nice relaxing place.