Taiwan is a fairly safe country to travel to, as far as health is concerned. It has little risk of the tropical diseases- cholera, malaria, dengue and the like- which are common in other Asian countries, although there is a minor risk of hepatitis (for which you should get vaccinations before you leave for Taiwan). Otherwise, simple precautions- like eating hygienic (and preferably freshly cooked) food, and drinking boiled/bottled water, should be enough. Fortunately, Taiwan’s healthcare system is pretty good, so even if you do fall ill, you should be in fairly safe hands. Get medical insurance done before you embark.
Taiwan’s a fairly safe place, as long as you keep a tight hold on your wallet while moving through crowded areas- pickpockets aren’t unknown. Don’t flash your money around, and keep your valuables locked, and you should be fine.