The Taipei subway is pretty new and clean. Fare just lower than $1 per trip but the network is not full. It will, however, probably get you to the main tourist places just fine.
By a chip at the pay stations. It looks like a poker chip. When you go through the turnstyles, just tap the chip in the area specified. This ‘logs you in’
When you exit the system, just drop the chip into the turnstyles to return it. This ‘logs you out’ and confirms you paid enogh when you bought the chip in the first place.
Slick system because the chip is easier to keep track of than the tickets in the Tokyo subway.