I am visiting Taiwan in the beginning of Jan, is it going to be real cold and less places to visit?

Comparing to other seasons in a year, Jan is truly the coldest time in a year. The average temperature is around 12 to 18C, but it will not affect your trip.

If you plan to visit some mountain areas such as the Yangming Mountain, you should prepare some thick coat even down jacket. By the way, it may rain a lot in winter time, so you are suggested to prepare the rain gear and keep attention on the local weather forecast to rearrange your trip.