Is it possible to head to Taiwan to learn Mandarin and look for work on the side, while on a student visa? Maybe someone may know or has gone through a similar experience.
It is not possible to learn Chinese. I’ve tried. But finding a job shouldn’t be too hard.
You’ll fail to learn Chinese and then you’ll get deported for working illegally on the student visa. 
(Okay, okay, you might learn some Chinese, maybe even a lot, before getting deported. Get a work visa, then study on the side. Eventually, marry a local and you won’t need the work visa.)
Sure. It’s an old trick for the non-degree-holders to get a student visa and then work illegally. Most of them never go to school but you could buck the trend. Half the people I know are working illegally and none of them ever have trouble. It seems that recently it’s become more of a pain to get a work visa than a student visa, so why not do it?
Haha maybe I should have been more clear. I graduate in December, but before going off to find work here in the states, I thought it would be fun to sharpen my Mandarin on the side (I am ABC, born and raised in CA). In the mean time I also thought it would be cool if I could find a job (teaching?) while I am on a student visa, and if my employer agrees, switch over to a working visa to allow an extended stay. Is this considered legal? Some countries don’t allow this so I just wanted some clarification before making drastic plans.
I plan to study at NTNU starting with the three month semester. I was staying in the Shida area last summer and liked it a lot. I want to continue that experience.