Taiwan shares several cultural taboos with other East Asian nations. Some Taiwanese are superstitious about anything connected with dying – unlucky things should never be mentioned. One thing to note is that the number 4 (four, pronounced 'si') sounds like the word for death in Mandarin.
In Taiwan culture, you should be mindful of certain things that are associated with death, such as do not write people's names in red. This has connotations of death. When writing someone's English name, this is not a problem, but avoid writing Chinese names in red.
Do not whistle or ring a bell at night. This is an "invitation to ghosts".
Do not point at cemeteries or graves. This is disrespectful to the dead.
There are numerous taboos dictating that certain objects shouldn't be given to others, often because the word for that object sounds like another unfortunate word:
Umbrellas - which in Mandarin sound the same as the word for "break up". Friends should therefore never give friends umbrellas. Instead, friends will euphemistically "rent" each other umbrellas for a tiny amount ($1, for example).
Clocks - The phrase "to give a clock" ("song zhong"), in Mandarin, has the same sound like the word "to perform last rites." If you do give someone a clock, the recipient may give you a coin in return to dispel the curse.
Shoes - Never ever offer shoes as a gift to old people, as it signifies sending them on their way to heaven. This is acceptable only if by mutual arrangement it is nominally sold, where the receiving party gives a small payment of about $10.
Knives or sharp objects - as they are made for or could be used to hurt the person.
The Taiwanese are certainly not puritanical and enjoy a drink, especially the locally brewed Taiwan Beer and Kaoliang. However, Taiwan does not have a culture of heavy drinking and is rare to see anyone drunk on the streets. While overindulging in alcohol is not a social taboo as such (and some people do so at weddings), it is considered a sign of lack of self-confidence and immaturity, and doing so certainly won't gain you any respect among Taiwanese friends.
It customary in Taiwan to remove your shoes before entering a house. You will find some slippers to be worn by visitors next to the entrance door.
It is likely to be the same ritual for bathrooms and balconies where you will be expected to remove your slippers to wear a pair of plastic sandals (though it is less shocking not to use the sandals by then).
As you will get along with Taiwanese people, you are very likely to receive small presents of any sorts. This will be drinks, food, little objects... These are a very convenient way to lubricate social relations for Taiwanese people and are specially commons between friends in their 20s. You should reply to any such presents with something similar, but it does not need to be immediate, or specific to the person (i.e. keep it simple).
As a teacher, you are not expected to offer anything in return (i.e. in a classroom environment) as long as the relationship stays formal. However, beware of the sometimes overly generous parents who can go as far as offering presents running in the thousands of NT$ and who will then expect you to take special care of their child (understand that their expectations will be considered as fair in Taiwanese culture).
You are not expected to tip in hotels, restaurants and taxis, though bellhops may still expect 50 TWD or so for carrying your luggage.
If you should need to use a temple's washroom, bow to any statues of deities you see on the way whether or not you believe in them. While most people will not mind you using the temple's washroom, they expect you to treat their place of worship with respect. If you plan to offer gifts (such as simple fruits) to the statues of deities in the temple, it is expected that you wash the fruits and your hands prior to offering.
In addition, upon entering and leaving a temple, do take note and avoid stepping on the extra step (a single raised step, similar to that of a stair's, often found at the gateways) that divides the outside and the inside of the temple. Always try to step over it instead of on it.
As with mainland China, symbols resembling backwards swastikas are commonly seen in homes and Buddhist temples. They are a Buddhist symbol and have no relationship to Nazism or anti-Semitism.
Taiwanese society is rather polarized by allegiance between supporters of the two major political blocks informally known as "Pan-Blue Coalition" and "Pan-Green Coalition", although there are large numbers of people who are either centrist or who don't care.
To simplify a very complex situation, pan-blue supporters tend to be more favourable toward the idea of (re)unification or maintaining a status-quo with China and pan-green supporters tend to be more favourable toward the idea of establishing a formally independent Republic of Taiwan, among other differences.
Although there are some correlations, it is highly unwise to assume anything about a particular persons political beliefs based on what you think you know about their background. Also, the very brief sketch of Taiwanese politics obscures a large amount of complexity.
Unless you know your listener well, it is unwise to say anything (either positive or negative) about the current government, about historical figures in Taiwanese history, about Taiwan's international relations, or about relations with mainland China. Some political figures such as Sun Yat-sen and Chiang Ching-Kuo are generally seen positively, but others (Chiang Kai-shek, Lee Teng-hui and Chen Shui-bian in particular) arouse very polarised feelings.
Some Taiwanese will get very offended if you imply that Taiwan is part of China. Others will get very offended if you imply that Taiwan is not part of China. Referring to the PRC as "mainland China" rather than simply China will tend not to offend anyone as the term is generally used to exclude Hong Kong and Macau as well, making it less subjective.
Referring to the Republic of China as a whole as "Taiwan Province" will draw a negative reaction from most Taiwanese. "Greater China" may be used in certain business contexts. Keep in mind, however, that there are so many subtleties and complexities here that if you are talking about these things, you've already wandered into a minefield.
However, simply referring to the island as 'Taiwan' is fine, as that is the name used by the locals, regardless of their political persuasion. Titles such as 'Republic of China' are reserved for official matters only.