If you want to work legally and full time in Taiwan, you will need to get an ARC.(Alien Resident Card).
An Alien Resident Card allows a foreigner teacher to work at up to two different establishments at the same time. The company that sponsors your ARC is listed on the card itself. An ARC lasts one year.
When you arrive in Taiwan, you can arrive via a visitor visa and change it to a working visa if you are hired by a school/company.
If you arrive on a Landing Visa, you most likely won’t be able to directly change into a work visa. Why? 1) Because you won’t have enough time. The process can take 10-14 days once all your paperwork is sent in. 2) The government may not view your Landing Visa as a transferrable one.
If you are hired in Taiwan, but your visa can’t be transferred, you will need to make a quick trip to Hong Kong with a letter from your employer (stamped with a chop) saying that you have been hired. In addition, you will need a letter for Foreign Affairs stating confirming this. Your employer should help you get this together. Hong Kong will then give you a standard visitor visa that is transferrable. You can then return to Taiwan and continue the ARC process. This whole process takes one day. Get to the Hong Kong office in the morning and pick it up in the afternoon.
If you want to work illegally in Taiwan, you can do it on a standard visitor visa or student visa. There are still short timers who come to Taiwan to earn a little money by subbing or filling in for short term jobs. These salaries are paid in cash and are good for some extra pocket cash.
Be aware, however, that if you are caught teaching illegally in a random check, you will be immediately expatriated and not allowed back into the country for 1-5 years. The school you are also teaching at is taking a risk hiring you as well. If caught with illegal workers, they will also be heavily fined and could lose their license.