Hey-ho! Here is a summary of places that you may want to visit in Taipei (including its outskirt attractions). Hope this helps!

Taipei / New Taipei City

1. Taipei 101
2. Chiang Kai Shek Memorial Hall
3. Sun Yat Sen Memorial Hall
4. Taipei Main Station
5. Xinbeitou (part 1, part 2)
6. Tamsui
7. National Palace Museum
4. Yongkang Street
5. Ximending

6. Night markets
    - Shilin Night Market
    - Raohe Night Market
    - Shida Night Market

7. Wufenpu
8. Maokong Gondola
9. Longshan Temple
10. Bopiliao Historical Site
11. Taipei Zoo
12. Wulai
13. Martyr Shrine
14. 228 Peace Memorial Park
15. 1914 Huashan Creative Park

Taipei's Outskirt

1. Shifen (part 1, part 2, part 3)
2. Jiufen
3. Yehliu Geopark (spring, summer)