Go to a 7-11 and grab yourself some local pineapple beer. I wasn't as big of a fan of the mango beer, but it indeed was quite tasty as well.
http://fountanabeverage.com/taiwan-beer/
There's more flavors?

Aww man, I missed out!
I ate so much street food. Bubble tea is nearly everywhere and reasonably cheap. I spent most of my time in Kaosiung and Kenting with locals, so I probably can't suggest much more than what I did above.
Pineapple cake
Steel (?) eggs
Tea eggs
Bubble tea
Are probably staples you can get anywhere that are reasonably priced and more "local".