Taipei has 2 citi-wide internet services. Taipei Free is totally free. WiFly is paid. Obviously, it is not as easy as 3G as you have to find the hotspots and can't move around while you are using it. 4G/LTE won't be available for a while.
Both are available in many MRT stations - although you have to find the optimal spot. Taipei Free is also available on some buses! WiFly is available in many Starbucks and 7/11s.
http://www.tpe-free.taipei.gov.tw/tp...Passanger.html tells you how to register.