Sometimes living in Taipei it seems like there is never really a good time. It often rains here for days and seems like it's never going to stop. The weather in Central and Southern Taiwan is generally better, at least in terms of more sunshine and less rain. Summers here are very hot and humid all over the island. Winters in the north can be surprisingly cool. Typhoons can hit the island any time from May to October.

In summary October and November are probably the best months to visit the north. Any time from October to March is probably a good time to visit other parts of Taiwan.

This webpage from the BBC gives a good overview of Taiwan's weather. The following quote comes from that page.

The whole island shares the tropical monsoon climate experienced on the southern Chinese mainland. Rainfall almost everywhere is over 2,000 mm/80in a year at low levels and much more in the mountains. More rain falls in the period May to September than in the rest of the year. Some of the heaviest falls of rain from July to September are brought by the typhoons of the South China Sea.

In winter, disturbed weather with cloud and rain affects the north and east coasts rather more than the south. Taipei not only has more rain in winter but on many more days.

Here are a couple more useful links. Wikipedia - Geography of Taiwan, Taiwan's Geography and Taiwan's Central Weather Bureau .