IMO the best way to get around in Taipei is subway, Uber, or Taxi 小黃. Almost any yellow car is a taxi, you can wave at them to get one.
Ningxia Night Market 寧夏夜市 Google Map Link
- From Zhongshan MRT station or Shuanglian MRT station, walks 10 min. Or take a taxi.
- This is where local people go, lots of good stuff.
Huaxi Street Night Market 華西街夜市 https://en.wikipedia.org/wiki/Snake_Alley_(Taipei)
- Famous for snake soup and other weird stuff
- Is next to Lungshan Temple listed below
PS. Most people recommend avoiding Shilin Night Market
淡水 https://en.wikipedia.org/wiki/Tamsui_District
- Easy to get to with subway
九份 https://en.wikipedia.org/wiki/Jiufen
- Take taxi or bus. Not sure if Uber still works in Taiwan or not
National Palace Museum 故宮 https://en.wikipedia.org/wiki/National_Palace_Museum (No subway)
Lungshan Temple 龍山寺 https://en.wikipedia.org/wiki/Lungshan_Temple_of_Manka (Has subway station)
- One of the oldest temple in Taiwan.
Liuzhangli 六張犁 Google map link (has subway station)
- Used to have lots of cemetery, including some famous people's. People say it's a good hike following Chonde Stree (崇德街) where you can see Taipei 101.
Treasure Hill 寶藏巖 https://en.wikipedia.org/wiki/Treasure_Hill
- Doesn't have treasure. It was an old settlement that has been turned into a tourist place.
Taipei 101
- Other than the building itself, the huge bookstore (Eslite) inside is worth checking out. It's like a public library.
Yangmingshan National Park 陽明山 https://en.wikipedia.org/wiki/Yangmingshan
- This is inside Taipei so it's easy to get to.
Taroko National Park 太魯閣 https://en.wikipedia.org/wiki/Taroko_National_Park
- In Hualien. Will take a day or 2 to get to from Taipei.