How to get a SIM card in Taiwan?

Here are the main 3 operators in Taiwan along with the frequencies they use, from https://guidetotaipei.com/article/cell-phones-and-sim-cards :

Regarding the location where to purchase raised prepaid SIM card, from http://prepaid-data-sim-card.wikia.com/wiki/Taiwan:

Purchasing at airports: There are kiosks located inside the terminals of both airports of all the major providers, that sell the tourist SIM cards. The cards are exclusively for tourists and at a slightly lower price then in the city, include more data and as well their WiFi access which provides many hotspots throughout the city for the duration of your purchased SIM cards validity. The only identification required for these is your passport.

• at Taoyuan International Airport (TPE): kiosks are located in both arrival and departure levels. Hours of operation: 8am - 10pm, daily.
• at Taipei Songshan Airport (TSA): kiosks are located in terminal 1 at international arrivals. Hours of operation: 6am - 10pm, daily.

Purchasing in the city: You can go to any of the provider stores in town to purchase a prepaid SIM card for regular prices. The difference with this option is that all operators except for Taiwan Mobile will require you to show a secondary piece of identification e.g. a drivers license and have to sign forms. Please note that in stores in the city employees are less likely to speak English as the staff at the airport. Unlike the SIM cards purchased at the airport however, those you buy in a store can be reloaded and the expiry date extended indefinitely as long as the period of time between recharges doesn't exceed six months. Bear also in mind that Taiwanese telecom regulations only allow one phone-number per person to be active at the same time. It will be difficult to check the status of the SIM when abroad, because none of the prepaid options offer international roaming, not even with text roaming only.

Opening hours at Taoyuan International Airport (TPE) airport (the main airport in Taiwan) can be consulted on http://www.taoyuan-airport.com/chinese/store3_1/1106. For example, for Chunghwa Telecom:

Some pictures from the Chunghwa Telecom in the Terminal 2 (they only accept cash by the way, but there is an ATM ~1 minute walking distance, just ask them where it is located):

Regarding the prices, for Chunghwa Telecom, from https://prepaidsim.twgate.net/index_en.html:

As you can see, data is cheaper than text messages / phone call, so you may want to use Line as most Taiwanese residents do.

At Taipei airport, after leaving the secure area and picking your baggage, there are numerous vendors of SIM cards. They offer different plans with 3G or 4G coverage and different packages, all advertised in large English signs. Most of them are at the edge of the terminal, turning left on exit from baggage claim and walking to the very end. By my calculation, pricing is very similar between them but it does depend on the mix of voice and data you are looking for.

The main operators, Chunghwa, Taiwan Mobile, FarEastone, all have a desk selling SIM cards, all located together in the Arrival hall. Can't miss them. They have different plans, although they tend to offer usually the priciest ones by default. You can ask for different options. I've used mostly Chunghwa so far, but they seem to be similar.

There can be quite long queues sometimes. While you're waiting in lines you can use one of the free Wifi networks.