Map from Google - Panama

In Panama the climate is tropical, hot all year round, with a dry season (January to April) and a rainy season (May to December).
The rains occur mostly in the form of heavy downpour or thunderstorm in the afternoon or evening, but they can continue even in the early hours of darkness.
The temperatures vary little throughout the year, however the cooler months, or rather the less warm, are usually October and November, at the end of the rainy season, although during this period the humidity makes the heat unpleasant. The hottest months are usually March and April, before the rainy season.

There are also some regions that have an equatorial climate, ie with no dry season: the north-west, near the border with Costa Rica, on the northern side of the mountain range called Cordillera de Talamanca, where the Amistad International Park is found, occupied by a cloud forest, and the south-east, namely the Darién Province, of the border with Colombia, where an impenetrable rainforest dominates, so much so that even the Pan-American Highway, which otherwise would connect the whole American continent from Alaska to Patagonia, is interrupted south of Yaviza (the interruption is called Darién Gap). In the former zone (see Changuinola), where about 100 inches of rain per year fall, there are two periods when the rains experience a relative decline: September-October and February-March, when from 4 to 6 inches per month fall, while the wettest months are July, September and December. In the latter zone, the Darién Province, where more than 120 in of rain per year fall, the only time when precipitation goes down to around 6 inches per month is January to March.

If we exclude the region of Darién, where it rains a lot on both sides, typically the Caribbean coast is rainier than the Pacific one, because the prevailing winds, especially in winter, are the north-east trade winds, which pick up moisture from the sea and then drop it on the mountain ranges that stretch through the interior of the country as a backbone, especially in the central-western region where they are higher.

In Panama City, the capital, located on the Pacific side, about 68 in of rain fall, almost all in the rainy season, from May to mid-December. This period is also muggy and has poor sunshine.
Here is the average precipitation.
Average precipitation - Panama City
Panama CityJanFebMarAprMayJunJulAugSepOctNovDecYear
Prec. (in)

In return, in the period from January to mid-April the sun shines and it almost never rains, but it's also hot, with a minimum around 73/75 °F and a maximum around 90 °F.
Here are the average temperatures of Panama City.
Average temperatures - Panama City
Panama CityJanFebMarAprMayJunJulAugSepOctNovDec
Min (°F)737373757575757573737373
Max (°F)889090908886868686848686

In the northern side, in Colón, in the Caribbean side of the Panama Canal, the rainfall patterns is the same, but the quantity is about twice as high, 140 inches per year. In the wettest month, November, the rains become monsoon-like, surpassing even 22 inches in a month. North-east of Colón we find the archipelago of the San Blas islands, coral islands inhabited by the Kuna Indians.
Here is the average precipitation in Colón.
Average precipitation - Colón
Prec. (in)3.721.84.914.213.41514.212.215.621.713.4132.1

The southern side, being leeward to the trade winds, is not only less rainy than the northern side, but also a bit hotter. On the south coast (see Panama City), the daily average temperature of the warmest month (April) is 82 °F, that of the cooler months (October and November) is 79 °F. On the contrary, along the north coast, the average ranges from 81 °F in April and May to 77 °F in November and December.

Panama City

To find some cool weather, you have to climb above 3,300 feet, in the so-called tierras templadas ("temperate lands"), which have a milder climate. In Boquete, 4,000 feet above sea level, on the slopes of the volcano Barú, which is the highest peak in Panama with its 11,400 feet, the average temperature fluctuates between 68.5 °F to 70.5 °F throughout the year.
Here are the average temperatures.
Average temperatures - Boquete
Min (°F)595961636363636361616159
Max (°F)797981817977797977777777

Above 6,500 feet, in the tierras frías ("cold lands"), nights can be cold, especially in the period from December to March. Above 10,000 feet, it can get cold even during the day.

Panama, southwest

Across the southern plains to the west of the Gulf of Panana (see David, Santiago), and in the Azuero Peninsula, the rains fall below 4 in already in December; during the dry season, the sun predominates along the coast and the heat in the interior, with highs around 95 °F in inland areas.
Here are the average temperatures in Santiago de Veraguas.
Average temperatures - Santiago
Min (°F)686868727272727272727070
Max (°F)909193939088888888868888

South of the coast, we find the pristine island of Coiba, which is also a national park, and a UNESCO World Heritage Site.
In the southern provinces of Herrera, Coclé and Los Santos, west of the Gulf of Panama (see Anton, Chitré, in the small area circled in red), precipitation drops to around 60 inches per year. In this region, the vegetation becomes less dense, so that there are also grasslands and shrubs. In the inland region there is even a desert, in the Sarigua National Park, actually due more to deforestation made by the last century's settlers to make room for pastures, than to climate.

The amount of sunshine in Panama is never great, because some cloudiness can always form; it's still better at the height of the dry season, from February to April. On the other hand, even in the rainy season, the sun makes its appearance on average for a few hours a day, at least in the plains and along the coasts. The cloudiest areas are the east, towards the border with Colombia, and the mountainous inland areas.

Panama is almost always spared from hurricanes, which pass at a higher latitude; the only hurricane to hit Panama in historical times was Martha, which followed an abnormal trajectory (from north to south) and hit the coast in Mosquito Bay in November 1969. However, the hurricane season runs from June to November, hence, it coincides with the rainy season, which typically tourists avoid.

The sea in Panama is warm enough for swimming all year round: the water temperature goes from 81 °F to 84 °F. And this applies to both coasts.
Sea temperature - Panama City
Panama CityJanFebMarAprMayJunJulAugSepOctNovDec
Sea (°F)818181818284828284848282

When to go

The best time to visit Panama, for a beach vacation or to visit cities and nature reserves, runs from January to mid-April, and in particular the months of February and March. It's hot almost everywhere, the sun shines and there is little rain, except in the north-west (province of Bocas del Toro) and in the south-east (province of Darién), which as mentioned have an equatorial climate, with frequent rainfall throughout the year.
In the south-western area circled in Bordeaux, you can go even in December.

What to pack

For plains and coasts: lightweight clothes, a light sweatshirt for the evening and air conditioning; in the rainy season (from May to mid-December in Panama City, and throughout the year in areas having an equatorial climate), a light raincoat or umbrella for thunderstorms. For the reef, equipment for snorkeling, water shoes or rubber soled shoes.
For tierras templadas: all year round, light clothing, a sweatshirt or sweater for the evening; raincoat or umbrella in the rainy season. In the dry season, sun hat and sunscreen.
For tierras frías: throughout the year, clothes for spring and autumn, t-shirt for the day, sweater or sweatshirt and jacket for the evening, raincoat or umbrella in the rainy season.
For high mountains: warm clothes, jacket, hiking boots.