All these routes I’ve tried myself so I can determine the best route to take and when to take it, be it walking, cycling or taking the cable car to the mountains.
Datong Walk

A walk best taken at the weekend that incorporates markets, gardens, the Fine Arts museum, temples, and other places of historical interest.
Zhongzheng Walk

A walk that includes the Chiang Kai-shek Memorial Hall, Performance Halls, a number of other historical buildings, the National Taiwan Museum and Peace Park.
Daan Walking Route

A walk to be taken at the weekend, that begins in a weekend only flower market, through the largest park in Taipei, Dongmen for food and finish off at the Chiang Kai-shek Memorial Hall.
Songshan/Xinyi Walking Route

A walk best taken at the weekend (although the week is fine too), you’ll visit the memorial hall, a great cultural park, then walk through the huge shopping district before finishing at Taipei 101.