It is free and a good exercise, and at the end of the hike you can see the best view of Taipei 101.
If your finding Taipei City to hectic and busy, you can easily head to one of the biggest parks I’ve ever seen in the world to relax. You can have a picnic or take a stroll through the tree lined trails.
Chiang Kai Shek Memorial－中正紀念堂
A classic Chinese style memorial of the past Taiwanese leader Chiang Kai Shek, it’s one of the biggest tourist attractions in Taiwan. However, despite this place being the one of the biggest tourist attractions in Taiwan it doesn’t feel all that overcrowded due to its immense size.
Jade and Flower Market －台北市建國假日花市
A must due when your in Taiwan, even if your not looking into buying any jade or flowers it is a nice way to spend the day window shopping and looking around.
The boardwalk at Danshui is a great way to spend the day or night relaxing and enjoying the oceanside walk, an added addition to the great walk is the street market with great street food and souvenirs. If you want you can also pay the same price to take the ferry across the ocean to the other side of the island, where more street food and window shopping abounds.
Ride alongside the River-
If you want some exercise and to have a relaxing day, you can take a ubike, or a cheaper more plentiful obikes or rent a bike at one of the nearby park areas, you can spend a good part of the day riding your bike along the many kilometers of paved bike lanes along the river experiencing one of the great parts of Taipei often overlooked.
Huashan 1914 Creative Park－華山1914文化創意產業園區
While not most of the exhibitions are paid to view, the rest of the creative park is free to stroll around and have a coffee or bite to eat.
A great place to window shop and get a quick bite to eat at a cafe or street food.
There are buses in the city available from Yuanshan MRT station that will take you up on the mountain, once on the mountain you can walk on any of the many hiking trails or have diner with a view at one of the many restaurants on the mountain. There are also many cafes you can have some coffee at, with a good view.
Another relaxing free thing to do in Taipei, right of the Brown line at Wende MRT station. After relaxing at the lake and watching some birds and old people, you can take a hike at the nearby hiking trail of the Bihu trail or Jinmianshan trail.
Let me know in the comments if you have any other free things to do in Taipei.