Map from Google - Taiwan

In the island of Taiwan, formerly Formosa, the climate is tropical, with hot and rainy summers, and mild winters. The period May-September is very rainy throughout the country, due to the summer monsoon, while in the other seasons, rainfall depends on latitude and slope exposure. However, winter is the least rainy season everywhere.
In winter, the temperature is pleasant in the south, where highs are about 22 °C (72 °F) in January, and the sun often shines, while in the north the climate is cooler and cloudier, with an average maximum temperature of 17/18 °C (63/64 °F). Due to both maritime influence and low latitude, night-time temperatures are rarely cold and never drop below freezing (0 °C or 32 °F).
In spring, in April and early May, early thunderstorms begin to occur in the afternoon, starting from the north and the interior. The monsoon arrives in the island around May 10 - May 20, coming from the south. It brings heavy rains throughout the island, but especially along the southwest coast, from Tainan to Hengchun, as well as in the highlands of the interior.
Summer is hot, humid and rainy, with more frequent rains (unlike in winter) along the southern coast. Temperatures are uniform from north to south: the daily average in the plains hovers around 28/29 °C (82/84 °F). The north-west coast, from Taichung to Taoyuan, experiences a relative decrease in rainfall in July. The amount of sunshine in summer is acceptable, because the tropical rains are usually intense but short-lived, however, in this season there's a sticky and unpleasant heat.
During summer, the island is often affected by typhoons, tropical cyclones able to bring strong winds and torrential rains; they typically cause heavier damage along the east coast. The period in which they are most frequent is from August to early October.
After the monsoon, in autumn, from October to December, the rains decrease and the muggy heat is replaced by a more pleasant weather. The only area which remains fairly rainy is the east coast, where more than 100 millimetres (4 inches) of rain per month fall still in autumn (see Hualien).

The capital Taipei is located in the northern part of the island, which is the rainiest along with the north-east. Here 2,100 mm (82 in) of rain per year fall. The rains are frequent from February to September, and definitely abundant from May to August. However, there is no dry season, and in winter the sky is often cloudy, because the winter monsoon, which is originally dry as it comes from the Asian mainland, picks up moisture over the sea and brings clouds, mists and some rain in the northern part of the island.
Here is the average precipitation.
Average precipitation - Taipei
TaipeiJanFebMarAprMayJunJulAugSepOctNovDecYear
TaipeiJFMAMJJASONDY
Prec. (mm)9014516518020532027026519011570752090
Prec.(in)3.55.76.57.18.112.610.610.47.54.52.8382.3
Days91312141213101210978129

The average temperature in the capital goes from 15 °C (59 °F) in January, to 29 °C (84 °F) in July and August. In summer, the "urban heat island effect", generated in the big city, makes the heat even more unpleasant, but fortunately the buildings are equipped with air-conditioning.
Here are the average temperatures of Taipei.
Average temperatures - Taipei
TaipeiJanFebMarAprMayJunJulAugSepOctNovDec
TaipeiJFMAMJJASOND
Min (°C)121315182124252523211714
Max (°C)181821252731333230262320
Min (°F)545559647075777773706357
Max (°F)646470778188919086797368

Taipei

As mentioned, in the south the winter is sunnier but also warmer, as we can see from the average temperatures of Tainan.
Average temperatures - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDec
TainanJFMAMJJASOND
Min (°C)121316192224252423211714
Max (°C)222326273030313131302623
Min (°F)545561667275777573706357
Max (°F)727379818686888888867973

In Tainan, on the south-west coast, the winter is dry and mostly sunny. Here is the average rainfall.
Average precipitation - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDecYear
TainanJFMAMJJASONDY
Prec. (mm)203550651703704304501553515151810
Prec.(in)0.81.422.66.714.616.917.76.11.40.60.671.3
Days4433101418171355399

The interior of the island is occupied by mountains, which reach their highest point in Yushan, 3,952 metres (12,966 feet) high, whose name means "Jade Mountain", due to the contrast created in winter between the snow-capped peak and the sky in the background. Above 2,000 metres (6,500 feet), winter is quite cold, and there's some chance of snow and frosts, while summer is cool but very rainy. On the top of Yushan, in the winter months the average temperature is around freezing, while in summer it's around 8 °C (46.5 °F). On average, even 3,000 mm (118 in) of rain or snow per year fall, with a remarkable maximum in summer, while snow is frequent in winter. Here are the average temperatures.
Average temperatures - Yushan
YushanJanFebMarAprMayJunJulAugSepOctNovDec
YushanJFMAMJJASOND
Min (°C)-5-4-203444420-3
Max (°C)3358101213131313106
Min (°F)232528323739393939363227
Max (°F)373741465054555555555043

The Penghu Islands (called Pescadores in the Portuguese era), located in the Taiwan Strait, are less rainy than the main island, since they receive about 1,000 mm (40 in) of rain per year.

The sea is warm enough for swimming all year round along the south coast, and from May to October along the north coast. In winter, however, the air temperature, though mild, is a bit low for beach life also in the south. Here is the sea temperature in Tainan, which as we said is located in the southwest.
Sea temperature - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDec
TainanJFMAMJJASOND
Sea (°C)232324262728292928272624
Sea (°F)737375798182848482817975

On the north coast, near Taipei, the sea temperature drops to 18 °C (64 °F) in January and February, and to 20 °C (68 °F) in the Pengchu islands.

When to go

The best time to visit Taiwan runs from November to March, being the mildest, and outside the monsoon rains, as well as the typhoon season. The east coast, however, is very rainy still in November.

What to pack

In winter: spring/autumn clothes, sweatshirt or sweater, light jacket, raincoat or umbrella. For the high mountains, down jacket, rain jacket, hat, gloves, scarf.
In summer: tropics-friendly, loose fitting clothing, a light sweatshirt and pashmina for air-conditioned places, light raincoat or umbrella.
For the reef, equipment for snorkeling, water shoes or rubber soled shoes.
For the high mountains, warm jacket, rain jacket, hiking boots.