A mix of Taiwanese (Minnan), Mandarin, Hakka and other Asian languages are spoken in Taiwan. Taiwanese is the largest group spoken by 70% of the population. In the North there is a segment of the people, known as "mainlanders", that mostly speak Mandarin. There is also a decent amount of Japanese spoken as there are many Japanese visitors.
All public announcements in the transportation system will be made in Mandarin, Taiwanese and Hakka, with the exception of the Matsu islands, where announcements are made in Mandarin and the Mindong dialect.
English is becoming increasingly useful, especially with people of the younger generations. There is an emphasis on English language education and compulsory courses are taught from mid elementary school onwards. However, attempts to speak Mandarin or Taiwanese are encouraged.
Some universities in Taiwan have programs promoting Chinese. These are offered to foreigners who wish to learn Mandarin Chinese. The writing system is usually traditional Chinese.
However, the Mandarin in Taiwan is different from the official Beijing Dialect. Taiwanese tend to use traditional Chinese characters, not the simplified versions used on the mainland. Taiwanese Mandarin also tends to not differentiate between the "S" and "Sh" sound. All people schooled after 1945 are generally fluent in Mandarin, although it is sometimes not their first language.
If you prefer to study at home, online classes may be perfect for you. There is also the ability to study online before you even enter Taiwan. One such school is www.studylanguages.org.