Note that there is difference between Mandarin that is spoken in Taiwan and the Mandarin that is spoken on mainland China (sort of like the difference between British and American English).    This is true especially for technical terms and idioms -- which developed separately since the 1949 civil war.  In terms of accents -- China has literally hundreds of local accents (not to mention local dialects)-- some virtually unrecognizable to the average Mandarin (ie... Beijing or Taipei standard) speaker.    Many Mandarin speakers claim to be able to distinguish an accent difference between Standard Taiwanese and Mainland Mandarin (ie... "can spot a Taiwanese after hearing a sentence"), but it is getting more and more difficult as the cultures intermix.  (note that standard Beijing Mandarin sounds very accented to Taiwanese, but is totally understandable).

Also note that this is complicated by fact that there is a separate local "Taiwanese Mandarin" accent and Taiwanese dialect spoken by Native Taiwanese (the ones with family roots in Taiwan from before the 1949 Chinese Civil war).  

It is very good to know some Mandarin in Taiwan -- you will endear yourself to the local populace if you try to speak to them.   But remember that most Taiwanese speak basic english and will have no problem understanding simple questions.  The drive to be an English speaking country (many TV hosts try to slip in a word or two of english just to add a touch of class to their progam:) has resulted in a large percentage of the younger population (starting in kindergarten) going to english "buxiban" (cram) schools -- so if a Taiwanese adult does not understand what you are saying, it is practically guaranteed that the group of 7 year-olds standing close-by are dying to translate for you! 

Putonghua - (literally translated "standard language") This is what Mainland Chinese call "Mandarin" 

Chung wen - (literally translated "central language") This is what Mainland Chinese AND Taiwanese call Mandarin

Guo Yue - (Literally - "country language" - but more accurately " the country's official language")  This is what older Taiwanese call Mandarin

Ni Hao or Ni Hao  Ma? - How are you? Hello.  

Nien Hao -  A formal, polite greeting reserved for seniors or bosses.

Xie, Xie (or sometimes romanized as hsieh, hsieh)  - Thank you (use this often, Taiwan people are very polite)

Bu Ke Chi - You are welcome

Mei yo wen ti - no problem (or don't worry)

Che So Jian - Bathroom

"Bu Hao E Tze" or "dway bu chee" - sorry, pardon.

One of the best online source for Mandarin phrases is :

  Chinese Tools: Phrasebook

This source uses the simplified Chinese characters used in mainland China, which is not easily understood by Chinese in Taiwan, Hong Kong, Macao and Singapore.