Map from Google - Bahamas

The archipelago of the Bahamas is stretched along a northwest-southeast axis: the northernmost islands are located to the east of Florida, while the southernmost ones are located to the northwest of Haiti. The climate is subtropical in the north, and fully tropical in the south, with a hotter and rainy season from late April to early November. The trade winds temper the heat in the summer months.
Proceeding from north-west to south-east, there's an increase in the winter temperatures and a decrease in the summer precipitation: this means that the north-western islands (Grand Bahama, Little Abaco, Andros and New Providence, where the capital Nassau is located) are cooler in winter and rainier in summer.
In fact, in Freeport, in the north-west, rainfall amounts to almost 1,500 millimetres (60 inches) per year, while it drops to 1,200 mm (47 in) in Nassau, 1,100 mm (43 in) in Eleuthera and San Salvador, 800 mm (31.5 in) in Mayaguana, and just 650 mm (25.5 in) in Great Inagua. In the south-eastern islands, not only the total amount of rainfall is quite scarce, but also there is a fairly significant decrease in the rains in July.
Here is the average precipitation in Nassau, located, as mentioned, in the northwestern island of New Providence.
Average precipitation - Nassau
NassauJanFebMarAprMayJunJulAugSepOctNovDecYear
NassauJFMAMJJASONDY
Prec. (mm)3545458011516015013516516585401220
Prec.(in)1.41.81.83.14.56.35.95.36.56.53.31.648
Days665611141616171599130

And here is the average precipitation in Matthew Town, in the arid southern island of Great Inagua.
Average precipitation - Matthew Town
Matthew TownJanFebMarAprMayJunJulAugSepOctNovDecYear
Matthew TownJFMAMJJASONDY
Prec. (mm)5035354085302050851207035660
Prec.(in)21.41.41.63.31.20.823.34.72.81.426
Days654564457108670

As for the temperatures, the main differences occur in winter, which is cooler in the northwest and warmer in the southeast, while the summer is hot and muggy everywhere.
Here are the average temperatures of Freeport, located in the island of Grand Bahama, the most northwestern of the archipelago, where temperatures resemble those of Miami, which is not far away.
Average temperatures - Freeport
FreeportJanFebMarAprMayJunJulAugSepOctNovDec
FreeportJFMAMJJASOND
Min (°C)161618192123242423211917
Max (°C)242426273031323232302725
Min (°F)616164667073757573706663
Max (°F)757579818688909090868177

In the coolest months (January and February), the average daily temperature is around 20 °C (68 °F) in the northernmost islands (Grand Bahama, Abaco), around 21 °C (70 °F) in New Providence and Eleuthera, 22.5 °C (72.5 °F) in San Salvador, 23 °C (73.5 °F) in Mayaguana, and 24.5 °C (76 °F) in Great Inagua. During winter, from December to March, cold air masses from the United States can lower the temperature for a few days, especially in the north-western islands, so that at night it can even get cold, below 10 °C in (50 °F) Freeport and Nassau; in addition, weather fronts can bring clouds and wind.
Here are the average temperatures in Nassau.
Average temperatures - Nassau
NassauJanFebMarAprMayJunJulAugSepOctNovDec
NassauJFMAMJJASOND
Min (°C)171718192123242424222118
Max (°C)262627283031323232302826
Min (°F)636364667073757575727064
Max (°F)797981828688909090868279

The temperatures in the warmest months (July and August) vary less from north to south, in fact they range from 27/28 °C (81/82 °F) in the northernmost islands, to 28/29 °C (82/84 °F) in the southern ones.
Here are the average temperatures of Abrahams Bay, in the southern island of Mayaguana.
Average temperatures - Abrahams Bay
Abrahams BayJanFebMarAprMayJunJulAugSepOctNovDec
Abrahams BayJFMAMJJASOND
Min (°C)202121222324252524242221
Max (°C)262727283031313231312827
Min (°F)687070727375777775757270
Max (°F)798181828688889088888281

The amount of sunshine in the Bahamas is good throughout the year, because the summer rains fall mainly in the form of showers and thunderstorms, in late afternoon or evening, so the sky doesn't remain overcast for a long time.
From June to November, however, the Bahamas can be affected by hurricanes, tropical cyclones of the Atlantic and the Caribbean, which are more likely from August to October.

Bahamas

The sea is warm enough for swimming all year round, even though it's slightly cooler in the northern islands, at least in the winter months: at Nassau the water temperature ranges from 24 °C (75 °F) in February and March, to 29 °C (84 °F) from July to September, while in the southern islands it does not drop below 26 °C (79 °F) even in winter.
Sea temperature - Nassau
NassauJanFebMarAprMayJunJulAugSepOctNovDec
NassauJFMAMJJASOND
Sea (°C)252424252628292929282726
Sea (°F)777575777982848484828179

When to go

In general, the best time to visit the Bahamas is the dry season, from late November to mid-April. However, in the north-western islands, from December to March the temperature is often not high enough for a beach holiday, and sometimes there are also some cool and windy days. In the southernmost islands, however, and in particular in Inagua and Great Inagua, winter is warm enough to enjoy the sun and the sea, while in the northernmost islands you may prefer April (the best) and May (still good but with the first summer rains).
In the rainy season, the heat is sweltering, and thunderstorms can be intense, even though the sun is still frequent. If you want to choose the summer, the month of July will be preferable, because there's a relative break in the rains (though not in Nassau and in the westernmost islands), and because hurricanes are less likely than in August.

What to pack

From November to April, in the northernmost islands: light clothing, but also a sweatshirt or sweater, a scarf for the breeze, jacket and umbrella; in the southern islands: light clothes, a sweatshirt for the evening, a scarf for the breeze.
From May to October: light tropical clothing, possibly a light sweatshirt and a light raincoat for thunderstorms.
For the reef, equipment for snorkeling, water shoes or rubber soled shoes.