What do they speak on the island of Taiwan? Is it Taiwanese? Or do they speak Mandarin Chinese (as many Americans call it)? The official language is Chinese. There is such a thing as Taiwanese that most of the population can understand and roughly half speak on a daily basis. A smaller portion of the island speaks what is known a Hakka. There are actually many more languages and dialects in Taiwan coming from the indigenous peoples who occupied Taiwan long before the Chinese speakers made their way over water to the island.
As you travel along the major highways from North to South you can hear specialty radio stations going to Hakka to Taiwanese. As you stop along the way in the cities and towns the local people also change from speaking Hakka to Taiwanese. The food choices and preparation styles change along with the scenery and the language as well.
Although the vast majority of people use Chinese as their language for business and socializing you do run into some elderly who don't really speak Chinese. I was surprised to find out that many of the older generation grew up speaking either Taiwanese or Hakka and never really learned Chinese properly. Sure they understand for the most part but getting them to speak it....? I have spent many hours in the same car with some older folks who spoke mostly in Taiwanese. That was an experience. Getting to know enough Chinese to enable me simple communication is tough enough without the added Taiwanese. But it was fun to learn a few phrases if for no other reason than to amuse them.
My experience in Taiwan has been surrounded by the Chinese, Taiwanese and Hakka languages. It just so happens that my girlfriend's parents are from two different parts of Taiwan. One from the South and the other from the North. Upon meeting family members from both sides I was quickly asked to memorize a short phrase in Hakka and Taiwanese, depending on which side we were meeting, and recite it to them. These little greetings were all in addition to the new Chinese that I was learning every day. It can be quite confusing as first but after hearing enough of each language I started picking up on the rhythm and intonation of them all. Not that I was easily identifying each one but that I could at least begin recognizing the phrases at times when another said it in a sentence during a conversation.
In conclusion, I would say that Chinese is by far the most useful language in general in Taiwan, regardless of where I went. Everybody can at least understand Chinese. Whether they speak it back to you or not is another matter. If you have the chance to learn some useful phrases from the locals from the North and the South, do it. Taiwanese and Hakka are quite prevalent in their respective areas. In most cases, as long as I was sincere in my approach, people were happy that I was trying to use their language.