Map from Google - Dominican_Republic

The Dominican Republic, the eastern part of the island of Hispaniola, has a tropical climate, hot all year round, with a dry season in winter, and a rainy season which goes from late April to October along the eastern and southern coasts, while the north coast, exposed to the trade winds, is rainy throughout the year, although it sees a relative lull in precipitation from June to September.
Thanks to the trade winds, that blow from the north-east from November to March, the north facing slopes are usually wetter than those exposed to the south. The presence of the mountain ranges add to the effects of the trade winds, so that some areas have a very humid climate and a lush vegetation, and others nearly barren.
For example in Azua, on the coast of the Ocoa Bay, just 640 millimetres of rain per year fall, and even less at Lake Enriquillo, closed between the two Sierras; even the south-west coast of the province of Pedernales is quite arid. On the contrary, in the capital, Santo Domingo, which lies on the southern coast but it's not protected by the mountains, about 1,400 millimetres per year fall.
Here is the average precipitation in Santo Domingo.
Average precipitation - Santo Domingo
Santo DomingoJanFebMarAprMayJunJulAugSepOctNovDecYear
Prec. (mm)65555570190140145175180185100851445

Here we find the distribution of the rainfall in Puerto Plata, on the north coast. Here the wettest months are even November and December, ie the first months in which the trade winds blow.
Average precipitation - Puerto Plata
Puerto PlataJanFebMarAprMayJunJulAugSepOctNovDecYear
Prec. (mm)175140130115130507070651102302301515

In the Dominican Republic, as generally in the Caribbean, the rains occur mainly as brief showers and thunderstorms, and are also often concentrated in certain periods, so it may not rain for an entire week even in the rainy season.

From June to November, the Dominican Republic can be affected by hurricanes, which, however, are most likely between August and October. The destructive hurricanes of category 4 or 5 are very rare, but when they directly hit a particular area, can cause considerable damage. Tropical storms, less intense than cyclones, are more frequent and hit the country on average once every two years.

The temperatures in the Dominican Republic are high throughout the year, at least in the plains: the maximum temperature ranges from 27/29 degrees Celsius in winter (if we can speak of winter), to 30/32 °C in summer, while the minimum ranges from 20/21 °C in winter to 23/24 °C in summer. During the summer season, from May to October, the heat is sweltering, while it is more pleasant, but still a little humid, in the winter.
Here are the average temperatures of Santo Domingo.
Average temperatures - Santo Domingo
Santo DomingoJanFebMarAprMayJunJulAugSepOctNovDec
Min (°C)202020212223232323222120
Max (°C)292930303031313231313130

In the mountainous areas, of course the temperature decreases with altitude. In Constanza, at 1,100 metres above sea level, during winter it's cool at night, with average temperatures around 9 °C, and sometimes can be cold, even near freezing, but the days are pleasantly warm, around 23 °C, while in summer the temperature ranges from 13 °C at night to 26 °C during the day.
If you want to climb the Cordillera Central, maybe up to Pico Duarte, which exceeds 3,000 metres, it's useful to remember that the temperature decreases by an average of about 6 degrees Celsius for every 1,000 metres.

Santo Domingo

For swimming, the sea is warm throughout the year: the temperature goes from 26 °C in February and March, to 29 °C between August and October, as we can see from the following table.
Sea temperature - Santo Domingo
Santo DomingoJanFebMarAprMayJunJulAugSepOctNovDec
Sea (°C)272626272728282929292827

When to go

In the inland valleys and along the south coast (see Santo Domingo), the best period goes from December to mid-April.
On the northern coast there is no real dry season, in fact the least rainy season is the period from June to September, which as in the rest of the country is sultry and has the danger of hurricanes; here you can choose the period from February to mid-April.

What to pack

From November to April: light clothes, a sweatshirt for the evening; for the mountains, jacket, sweater and hiking boots.
From May to October, lightweight clothing, possibly a light sweatshirt and a light raincoat for thunderstorms; jacket, sweatshirt and hiking boots for the mountains.