I know that after you have been in Taiwan for a number of years that you can apply for an APRC which allows you to work anywhere and would not require you to get individual work permits for each school.
Now does anyone know how many years you have to be in Taiwan to get one of these? Or any other info on it?
cheers 
You need to be intaiwan for 7 years running on an unbroken ARC, and then you can apply for the open work permit (it’s is not automatic.). You have to apply for that seperatly after you get your APRC.
There is a lot of very good info in the visas and work forums. Also check out iff.npa.gov.tw/enfront/index.php Good luck! 