The Best place to check for the current   weather situation in Taiwan before coming over for a trip is:

Central Weather Bureau 

Winter  

Chinese New Year's and the two weeks following are generally a crush of people on vacation from business and schools. 

Summer

Summer months are generally very subtropical hot and humid (especially July and August) throughout day and night.

However, many fruits will be in season and at the peak of deliciousness, like lychees.

Snacks like crushed ice "bao bing" (泡冰) will be common and refreshing.

Typhoons (Hurricanes)

Typhoon season sees many storms from mid-summer through early December.

Earthquakes

Minor, almost imperceptible, earthquakes are common throughout the year.  Most are epicentered off the coast in deep, seismically active underwater locations.