Taipei metro is extensive, cheap, and is really easy to use
Most trips within the city will cost you less than 35 NT ($1.2 USD). Minimum fare is 20 NT (65 cents US) and maximum fare is 65 NT ($2.1)
First and last train times vary by station, but in general the system runs between 6am and 11:30pm
How to buy tickets:
The metro sell three main kinds of tickets - single trip, 1-3 day passes, and EasyCard. The best option is EasyCard, which can be used not only on the metro, but also buses, airport train, taxi, and even short distance TRA trains. 1-3 day passes are less useful because it's only limited to the metro and is only economical if you plan to make more than 6 metro trips per day.
To buy single trip tickets, you can use the automated bilingual vending machines at every station. Unlike many metro system elsewhere in the world, on these machines you don't pick the starting and ending station. Instead, the screen will display the fare to every other station from where you are, and then you simply pick on the screen how much fare you want to purchase (20NT - 65 NT). Check this video:
To buy EasyCard, you can use the dedicated EasyCard machines in every metro station. On them you can buy, load, and return cards for refund. You need to load a minimum of 500NT to get an EasyCard, which includes 100NT in deposit and 400NT in usable values. Check this video:
Buying tickets - single, passes, or EasyCard - is with cash only, no credit card is accepted for these. Also, single trip ticket machines will give you change, but EasyCard machines will not give you change and load the entire amount into your card.
You get a circular plastic token if you get a single trip ticket. EasyCard is the standard plastic card. You tap the token or the card to get into the station. Hang onto the token, you will need to return it when you exit later. With EasyCard also need tap out of the station.
Best way to plan your trip with metro is actually Google Maps, which from my experience has very up-to-date line information. It's also easy to see on the maps how to get to the metro station from where you are or to your final destination.