If you like food (street food), you have to go to the night markets!
In Taipei, I would recommend - Shilin Night Market and Raohe Night Market.

If you like seafood, you should try Addiction Aquatic Development. It is a seafood market with very fresh seafood, very affordable! There are some restaurants (mostly standing).

I also recommend to take a day trip to Jiufen (Spirited Away inspired town).

For shopping, I recommend to go to Dongqu (忠孝 area). There are nice boutiques and all walkable.