If yo got free time in Taiwan, you do not want to just hang up, next to a 7/11, and drink some Taiwan beer. The way people here use their free time differs slightly from many western countries especially those from Europe like France, Italy, Germany or Spain. Whereas most people from these countries understand relaxing means doing nothing, here in Taiwan there is often a complete different approach.
Dancing for example, most Americans do any move that comes into their mind and enjoy doing it without being very professional about it . In Taipei there are many gyms and dance-clubs offering dancing lessons for pretty every style. These courses are busy filled with very ambitious participants.
There are still things you can do if you feel like being lazy. Go to a spa, get a foot massage find a nice place at the riverside park and read a book.
If your accent is set on activities, then you will prefer riding your bike on one of those endless biking trails. Maybe going to the gym or doing some sight seeing.