In my (limited) tourist experience, unless you want to ride a C-bike in Kaohsiung, you can stick with the Easycard bought at Taoyuan airport's MRT station for all of your (public, local) transportation needs.
There are several stored value cards available from different providers around the country, but the most important one are Taipei Easycard and Kaohsiung's iPASS. Fortunately they are mostly interoperable and both of them will allows you to pay on every metro and local buses (at least those with card readers, some in smallish towns still require cash), long distance buses and some (but not all) trains.
They will also be accepted on some shop (and can be reloaded in the main convenience store chains), although their acceptance is not as unibersal as the octopus card in Hong Kong, for example.
They also allows you to use (and pay for) the (city operated) bike sharing systems with one caveat: Kaohsiung's C-bike only accept iPass, not the easycard, while the Youbike (in Taipei or Taichung, among other) systems accept both.