Taiwan's national language is Pǔtōnghuà (Chinese as spoken in mainland China); the written form uses traditional characters (as does Hong Kong). Other Chinese dialects you will commonly hear are Hoklo and Hakka.
English is widely spoken, especially by young people and in cities. Road signs and announcements on trains and buses are also also given in English. However, many restaurants only have menus in Chinese.
Taiwanese are very friendly and if you are having trouble communicating, a passerby will usually come and help translate.
Taiwan is a fantastic place to learn Chinese, and there are many universities and private schools that offer courses. Most offer classes for two to four hours a day, five days a week, as well as private classes for as many hours as you like. Costs vary from NT$500 to NT$700 for a private one-hour class and around US$4000 per semester at a top university program.
Study in Taiwan (www.studyintaiwan.org) is an excellent resource for finding schools and how to apply for a scholarship and study visa. The Ministry of Education (http://english.moe.gov.tw/) is another good site to get up-to-date information.
Some of the better-known programs include the International Chinese Language Program at National Taiwan University and the Mandarin Training Centre at National Taiwan Normal University. Both universities are in Taipei but there are programs around the country.
A decent private school if you are looking for a less rigorous program is the Taipei Language Institute.