These two time periods are considered to be the best times to visit Taiwan. Between April and May, the weather is warm and not at all humid, making it conducive for travelling around. Another reason is that the festival of the goddess Mazu's birthday is celebrated in April and this is a great time to be in Taiwan. Between October and December, the weather gets chilly and it does not even rain much.
More about Best Time to Travel to Taiwan
June - August (Summer)
Taiwan summers are both hot and humid. It is not at all a good time to visit the country. Tropical cyclones also occur during this season.
March - May (Spring)
It does tend to rain around this time. However, this is the best kind of weather for travelling purposes, as it is pleasantly warm.
December - February (Winter)
Winters here are chilly and dry. It does not usually rain during this time. This weather is perfect for walking around the city and taking in its beauty, without worrying about the heat.