For North America and Eurozone countries, a 90-day visa on arrival is given right when you get off the airplane.
Beyond that, to work in Taiwan you will need an "Alien Resident Card" or ARC granted to teachers and white-collar workers. This means you can come and go as you please, get access to Taiwan's stellar public health care, and generally never need to worry about visa matters so long as you're employed.
If you have an ARC for five years consecutively, you can apply for an APRC, or as you guessed, the P stands for Permanent residence card. This means you're golden and you can live and work in Taiwan just like the locals do. No more worrying about your visa if you quit or lose your job, and no employment restrictions whatsoever. You can even work in a 7-Eleven or as a garbage man.
Now the holy grail of visas is no visa at all, meaning citizenship. I've heard there are only about 100 foreigners who've attained Taiwanese citizenship, with Taiwan being the closed country that it is. And these are foreigners who have likely lived in Taiwan for twenty plus years, invested in real estate or business, or have done something remarkable.