Taichung 台中
A big-city stop between the northern hills and the central mountains.
Although a bit far out of the city, the bicycle historian in you may like the Cycling Culture Museum 自行車文化探索館 on the site of the huge Giant-owned plant. STAND Collection's premium apparel store would be the right place if you've found out your current bib isn't (good) enough.
Taichung, like every Taiwanese city, has good quality night markets; Fengjia 逢甲夜市 in the northwest is the most famous one, but you should be able to find closer ones. At the end of the day we found ourselves very happy at the Taihu Brewing 臺虎精釀 rooftop bar/restaurant for a relaxing night with a pint. For our morning coffee we found the small and cute Tacit Cafe close by, nicely located in a busy market street.