What are the recommendations or any suggestions to travel on the early September 2016 to Taiwan?

Generally, it can be fine to travel during that time, with comfortable temperature (26C-30C) and frequent but light rainfall. Taipei, Hualien and Kenting are recommended destinations. Do prepare enough summer clothing, very light jacket (maybe cold with indoor air-conditioning) and rain gear. However, it shall be during typhoon season, so there is possibility of typhoon, which may ruin your trip. So it is necessary to check the forecast in advance. I just this website presents accurate data, but only for recent several days.