Here is the part where you NEED to pay attention. Stop, sit up, read carefully. This will save you headaches, money and frustration.

Americans, Australians, British, Canadians, Irish and New Zealanders can enter Taiwan without getting a visa before hand.  South Africans cannot get a visa on entry.

Visas given at the Taiwan airport on entry are called "Visa Exempt Entry".

Here is the important part, if you enter just by rocking up at the airport in Taiwan, the visa they will give you will only be good for 30 days, it is not extendable AND your visa CANNOT be converted to a visitor or resident visa. If this is the case, you have to LEAVE Taiwan and re-enter. Expensive. Time-consuming. Frustrating.

Teachers entering Taiwan to teach in public schools follow the official format for getting visas. These teachers will apply for a Visitor Visa for Employment Purpose.

Teachers entering Taiwan to teach in private language schools (buxibans), private schools, or international schools, will most often enter on Visitor Visas for Tourism or Visiting Purposes and they will change the status of their visas once they are in Taiwan. 

All applications for Residency Visas (also known as work visas or ARC Alien Residence Cards) are made once you are in Taiwan. This is generally why private language schools and private schools choose to bring you in on a Visitor Visa for Tourism purposes. It saves them having to do more paperwork and jump through more hoops.