Is there a universal transportation card that is valid for MTR and buses all over Taiwan? Or do I have to buy a new card each time I visit a new city in Taiwan?
If there is no universal card, I see that there is an Easy card in Taipei. What are its analogues in Taichung and Kaohsiung, and can I buy them at THSR stations in each of the cities on arrival?
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 ones are Taipei Easycard and Kaohsiung's iPASS. Fortunately they are mostly interoperable and both of them will allow you to pay on all metros 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 at some shops (and can be reloaded in the main convenience store chains), although their acceptance is not as universal as the Octopus card in Hong Kong, for example.
They also allow you to use (and pay for) the (city operated) bike sharing systems with one caveat: Kaohsiung's C-bike only accepts iPass, not the easycard, while the Youbike (in Taipei or Taichung, among other) systems accept both.
