Map from Google - Papua_New_Guinea

The climate in Papua New Guinea is hot and humid throughout the year along the coasts and plains, while it's cool and progressively colder as you go up in altitude. In much of the country, covered with dense rainforest, there is not a dry season, so we can speak of equatorial climate, while in some inland valleys and along the south coast, there is a relatively dry season from July to September.
The dominant currents are related to the two monsoons, the north-western one from December to April, and the south-eastern one from May to October, which generally bring rains in the slopes exposed, while many areas receive rainfall from both. Rainfall varies typically from 80 to 160 inches per year, with some higher peaks on the exposed slopes, where it also reaches 23/26 feet per year, while it drops to 40/60 inches in the southern coast, overlooking the Gulf of Papua and the Coral Sea.

Papua New Guinea

In Madang, on the north-eastern coast of New Guinea, 140 in of rain per year fall, with a maximum of 17 inches in April and a minimum of 4.7 inches in August. Here is the average precipitation.
Average precipitation - Madang
MadangJanFebMarAprMayJunJulAugSepOctNovDecYear
MadangJFMAMJJASONDY
Prec. (in)1211.81516.915.210.87.74.75.71013.414.6137.8
Days17161918171111910111318170

East of the mainland, even the islands (including New Ireland, New Britain and Bougainville) are very rainy throughout the year. In general, the rains are most abundant from December to April, but not everywhere: in Lae, facing south-east in the Huon Gulf, the rains are most abundant in July and August, when more than 20 inches per month fall.
The temperatures are high and stable throughout the year in the north, around 86/90 °F during the day, while in the south, located farther from the equator, they decrease a bit in the period that can be called winter, from June to September.
Here are the average temperatures of Madang.
Average temperatures - Madang
MadangJanFebMarAprMayJunJulAugSepOctNovDec
MadangJFMAMJJASOND
Min (°F)737373737372727272727373
Max (°F)868686868686848486868686

The amount of sunshine is usually low or just sufficient in a large part of the country, even poor in the mountainous inland regions, where the sky is often cloudy, while along the southern coast, in the dry season from May to October the sun often shines.
The capital Port Moresby is located in the southern area, which as mentioned is less rainy and more sheltered, so that it receives only 40 in of rain per year, with relatively dry period from May to November, when they drop below 2.8 inches per month; in particular, very little rain occurs from June to October. Here is the average precipitation.
Average precipitation - Port Moresby
Port MoresbyJanFebMarAprMayJunJulAugSepOctNovDecYear
Port MoresbyJFMAMJJASONDY
Prec. (in)7.17.76.74.12.61.41.20.711.424.340.1
Days981063432234761

Here, as mentioned, the amount of sunshine is higher than other areas of the country. In the capital, the period from June to September is the best because it is also, albeit slightly, the least hot, with maximum temperatures around 84/86 °F, instead of 88/90 °F which are recorded in the rest of the year.
Average temperatures - Port Moresby
Port MoresbyJanFebMarAprMayJunJulAugSepOctNovDec
Port MoresbyJFMAMJJASOND
Min (°F)737272727272707072727273
Max (°F)888888888686848686889090

The sea is always warm enough for swimming throughout the country, although in the southernmost part, where lies the capital, becomes a bit less warm from July to October, as we can see in the following table.
Sea temperature - Port Moresby
Port MoresbyJanFebMarAprMayJunJulAugSepOctNovDec
Port MoresbyJFMAMJJASOND
Sea (°F)848484848281797779798284

The interior of the mainland (eastern New Guinea) is crossed by mountains: in the north the Victor Emanuel Range and the Bismarck Mountains, the latter culminating in the 14,793-feet high Mount Wilhelm, while in the south there are the Owen Stanley Mountains, culminating in Mount Victoria, 13,248 feet high. At 5,000 feet the temperature is pleasant: cool at night, around 59 °F, and warm during the day, around 77 °F. Above 11,500 feet, the forest gives way to a sparse mountain vegetation: here the cold is constant and sometimes it may even snow.
From the mountains, several rivers flow, such as Sepik and Fly, which in lowland areas, often at great distance from the mouth, give rise to marshes and swamps, rich in mangroves and gallery forests.

Papua New Guinea can be affected by tropical cyclones, which in the South Pacific occur from November to mid-May, although they are more likely from November to March. The part of the country that is directly affected is the south-central. The map below is an indication of the areas that have been affected in the past. The northern areas, however, may be partly affected by an increase in wind, clouds and storm surges.

Papua New Guinea, area affected by cyclones

The climate of Papua New Guinea is influenced by the so called ENSO cycle: in the years of El Niño, the rainy season is warmer and drier than normal, and the monsoon arrives often late, while the dry season is cooler than normal in the south; in La Niña years, the rainy season from December to April is rainier than normal and can lead to flooding, while the dry season is warmer than normal in the south. During neutral phases (neither El Niño nor La Niña), however, it's more likely for cyclones to form.

When to go

The best time to visit Papua New Guinea, at least the southern coast which includes Port Moresby and is the only one which experiences a real dry season, runs from June to September, because it is also the coolest. Even May, October and November along this coast have little rains, but they are a bit warmer.
On the contrary, as we have seen the other parts of the country do not experience a real dry season, in fact they are rainy all year round, and indeed some areas, such as the coast where Lae is located, receive the maximum rainfall in this period. However, from June to September it's the best time to visit the country as a whole, recalling that there may be showers and thunderstorms, more or less intense, in most of the country.

What to pack

All year round, tropics-friendly, loose fitting clothing, sun hat, a scarf for the breeze, a light sweatshirt for the evening, light raincoat or umbrella. For the reef, equipment for snorkeling, water shoes or rubber soled shoes.
In inland areas, around 6,500 feet, spring/autumn clothes, sweater and jacket for the evening, umbrella. For the highest peaks, warm clothes, down jacket, gloves, hat, hiking boots.