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 23 °C (73 °F) in January, and the sun often shines, while in the north (see Taipei) the climate is cooler and cloudier, with an average maximum temperature about 18 °C (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). However, sometimes in the north there can be very cool days, with highs about 10/12 °C (50/54 °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: in July and August, the daily average in the plains hovers around 29/30 °C (84/86 °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, and in winter is the coolest. The average temperature goes from 16 °C (61 °F) in January, to 30 °C (86 °F) in July. 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)141416202325272625232016
Max (°C)181921252831333330272420
Min (°F)575761687377817977736861
Max (°F)646670778288919186817568

Here 2,400 mm (95 in) of rain per year fall, with a peak in the hottest period, from June to September. Here is the average precipitation.
Average precipitation - Taipei
TaipeiJanFebMarAprMayJunJulAugSepOctNovDecYear
TaipeiJFMAMJJASONDY
Prec. (mm)8517018018023432524532036015085752405
Prec.(in)3.36.77.17.19.212.89.612.614.25.93.3394.7
Days141516151516121414121212166

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. Even in spring the sun can be seen rarely. Overall Taipei is a dull city: in a year there are only 1,400 hours of sunshine. Here are the average sunshine hours.
Sunshine - Taipei
TaipeiJanFebMarAprMayJunJulAugSepOctNovDec
TaipeiJFMAMJJASOND
Sun (hours)333344665433

Along the northern coast, the sea is warm enough for swimming from May to October, while in winter it becomes very cool, as we can see from the sea temperature at Keelung, about 20 kilometres (12 miles) northeast of Taipei.
Sea temperature - Keelung
KeelungJanFebMarAprMayJunJulAugSepOctNovDec
KeelungJFMAMJJASOND
Sea (°C)181819222527282827252220
Sea (°F)646466727781828281777268

Taipei

As mentioned, in the south the winter warmer, as we can see from the average temperatures of Tainan.
Average temperatures - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDec
TainanJFMAMJJASOND
Min (°C)141518212426262625232016
Max (°C)232426293132333232312824
Min (°F)575964707579797977736861
Max (°F)737579848890919090888275

In Tainan, on the south-west coast, 1,700 mm (67 in) of rain per year fall, and the dry season is long, from October to March, while showers begin to increase in April. Here is the average rainfall.
Average precipitation - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDecYear
TainanJFMAMJJASONDY
Prec. (mm)153040801753703603951803015151700
Prec.(in)0.61.21.63.16.914.614.215.67.11.20.60.666.9
Days455791312151022387

In Tainan, and in general in the south, the winter, in addition to being drier, is also more sunny than in Taipei. In Tainan there are 2,200 hours of sunshine per year.
Sunshine - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDec
TainanJFMAMJJASOND
Sun (hours)666666766666

The sea is warm enough for swimming all year round along the south 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.
Sea temperature - Tainan
TainanJanFebMarAprMayJunJulAugSepOctNovDec
TainanJFMAMJJASOND
Sea (°C)232324262728292928272624
Sea (°F)737375798182848482817975

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.

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.