Greeting is one of the best entry to a language and culture (damning, the other, in my opinion), and even between Taiwanese Mandarin and Chinese Mandarin, there are some differences. For example, the well-known greeting "你好" (pronounced as "ni hao," meaning "may you be fine") is not so usually used in Taiwan Mandarin. It just... sounds a little weird to native speakers.

One reason is plausible: in Asian culture, including Japanese and Ancient Chinese, to mention people by their titles is thought much more respectful than by saying "you". We are even taught not to say "may I ask you for it, sir?" but "may I ask sir for it?" as if you have no any idea that the sir you are talking to is in front of you. Therefore, we always say "老師好" ("lao shi hao", "may teacher be fine) for greeting to a teacher, "叔叔好"/ "阿姨好" ("shu shu hao"/ "a yi hao") to a male/ female elder, "X 先生/小姐好" ("X xian shen/ xiao jie hao", in which X is his/her family name) to a man/ woman. (Notice: it is rude to call a woman "小姐" ["xiao jie"] in China, which insinuates prostitution; "女士" ["nü shi"] is suggested instead, sounding too official and stilted to Taiwanese.)

The other reason we are unfamiliar with "你好" ("ni hao") is a little implicit and subtle: "好" ("hao") is a pretty vulgar word, and instead, "安" ("an," meaning "pacific") is more cultivated. In most of archaic usage, Chinese say "問安" ("wen an") instead of "問好" ("wen hao") to describe the action of greeting. In the end of a formal letter, we write "恭請 X安" ("gong qing X an," X dependent on the relation between the reader and the writer, meaning "pleading for being pacific"), not "恭請 X好." Before "你好" popularized, we even translated "hello," "salute," or "こんにちは" into "日安" ("zhi an," meaning "good day"), which is an obsolete word now. Therefore, in Taiwanese Mandarin we say "早安" ("zao an") instead of "早上好" ("zao shang hao") for "good morning," "午安" ("wu an") instead "中午好" ("zhung wu hao") for "good afternoon," and "晚安" (wan an) instead of "晚上好" (wan shang hao) for "good evening." By the way, you may say just "早" ("zhao") for "good morning" both in Taiwanese and Chinese Mandarin, but you CANNOT say "午" or "晚" for "good afternoon" or "good evening" anywhere; I don't know why but that's it.

So, you may ask, what is the proper greeting for all situation in Taiwanese Mandarin, akin to "你好" in Chinese Mandarin? I am sorry that there is no such a word. But, on the other hand, fortunately, on this cosmopolitan insula, people are used to saying "hello" (pronounced as "HA-lo"), "hi," and "hey" for daily greeting. If you want to draw attention from someone you haven't met, "umm..." with a confusing smile is a good manner, just like "あの..." in Japan, or you may say straightforwardly "不好意思" ("bu hao yi si") for "excuse me," making you sound amazingly native!