Map from Google - Burma

In Burma (or Myanmar), the climate is generally tropical, with a rainy season from mid-May to October due to the southwest monsoon, a cool and dry season from November to mid-February, and a hot pre-monsoon season from mid-February to the beginning or the middle of May.
In addition, there are mountainous areas, with a mild or cool climate, but also a portion of the Himalayan in the far north, which is very cold, at least at hith altitude.
The rains are very abundant along the coast, and are usually abundant in the mountain slopes, while some inland valleys and sheltered plains are almost dry.

Climates in Burma

We can distinguish among the following climatic zones:
1 - In the far north, a mountainous area, offshoot of the Himalayas, with perennial snows above 4,500 metres (14,800 ft). The highest mountain is Hkakabo Razi, 5,881 metres (19,295 feet) high. To the south, at the foothills of the mountains, the rains are abundant, given that they exceed 3,000 millimetres (118 inches) per year, with heavy rainfall from May to October, and a dry season in the rest of the year.
2 - Two mountain ranges in the west and the east (the Arakan Mountains and the Shan Plateau), with a mild climate. In the monsoon season, the rains are abundant, torrential at times, in the western range, while they are moderate in the eastern range, around 1,500 mm (60 in) per year. In winter, nights are cold.
Here are the average temperatures of Taunggy, located in the east, at 1,400 metres (4,600 feet) above sea level.
Average temperatures - Taunggy
TaunggyJanFebMarAprMayJunJulAugSepOctNovDec
TaunggyJFMAMJJASOND
Min (°C)781115171717171716118
Max (°C)222427282725232424232321
Min (°F)454652596363636363615246
Max (°F)727581828177737575737370

3 - A central area, which includes the capital, and has a tropical climate, a bit cool in winter in the north, with heavy rains, but not exceptional, during the monsoon season. In the northern part, winter is quite cool, as we can see from the average temperatures of Myitkyina.
Average temperatures - Myitktyina
MyitktyinaJanFebMarAprMayJunJulAugSepOctNovDec
MyitktyinaJFMAMJJASOND
Min (°C)101216202223242423211611
Max (°C)232530323231303031302724
Min (°F)505461687273757573706152
Max (°F)737786909088868688868175

4 - An inland area, sheltered from the monsoon, therefore almost barren (see Mandalay), which gets very hot in the pre-monsoon season.
Here are the average temperatures in Mandalay.
Average temperatures - Mandalay
MandalayJanFebMarAprMayJunJulAugSepOctNovDec
MandalayJFMAMJJASOND
Min (°C)131520242626262525241915
Max (°C)293236383734343233323028
Min (°F)555968757979797777756659
Max (°F)8490971009993939091908682

5 - Along the coast, an area with a warm climate all year round (but a little cool in December and January in the northern part facing the Bay of Bengal), with very heavy summer rains.
Here are the average temperatures of Sittwe, in the northern part of the coast.
Average temperatures - Sittwe
SittweJanFebMarAprMayJunJulAugSepOctNovDec
SittweJFMAMJJASOND
Min (°C)151618242525252525242117
Max (°C)282931343230292930313029
Min (°F)596164757777777777757063
Max (°F)828488939086848486888684

In winter, from mid-November to mid-February, the climate is relatively cool, especially at night and in the centre-north. The weather is mostly sunny and warm in the daytime, but sometimes cool air masses from China may lower the night temperatures almost to the freezing point in the north, about 5 °C (41 °F) in the centre-north, 8/10 °C (46/50 °F) in Yangon and on the west coast, and 12/15 °C (54/59 °F) along the south coast.
In winter, the biggest differences are recorded: the average daily temperature in January is 16 °C (61 °F) in Myitkyina, 20 °C (68 °F) in Mandalay, 21 °C (70 °F) in Sittwe, which is located on the northern coast of the Bay of Bengal, 25 °C (77 °F) in Yangon and 26 °C (79 °F) along the south coast. So it's hot even in winter only from the area of Yangon (the former Rangoon), the capital, and the nearby beaches of Chaungtha and Ngwesaung.
Average temperatures - Yangon
YangonJanFebMarAprMayJunJulAugSepOctNovDec
YangonJFMAMJJASOND
Min (°C)181922242525242424242219
Max (°C)323536373330303030323232
Min (°F)646672757777757575757266
Max (°F)909597999186868686909090

In the pre-monsoon period, from mid-February to the beginning or middle of May, the weather remains sunny, but the temperature rises gradually, and in mountainous areas there can be some atmospheric instability in the afternoon, with the first thunderstorms. It's the hottest time of the year throughout Burma, especially in valleys and inland plains, where the heat becomes unbearable, and people cannot wait for the arrival of the monsoon. The temperature can reach 37/38 °C (99/100 °F) along the southern coast, 40 °C (104 °F) in Yangon, and even 43/45 °C in Mandalay and in inland areas. Typically, April is the hottest month, but the first part of May, before the arrival of the monsoon, is similar.
On the coasts of the Bay of Bengal and the Andaman Sea, the pre-monsoon heating is less intense (both because of the thermal inertia of the sea, and because the monsoon rains arrive earlier), as we can see from the temperatures of Mergui.
Average temperatures - Mergui
MerguiJanFebMarAprMayJunJulAugSepOctNovDec
MerguiJFMAMJJASOND
Min (°C)202122232323222222222120
Max (°C)303132323129282828303030
Min (°F)687072737373727272727068
Max (°F)868890908884828282868686

The monsoon season is less hot: the maximum temperature goes down to around 28/29 °C (82/84 °F) on the coast, 30 °C (86 °F) in Yangon, and 33/34 °C (91/93 °F) in Mandalay, but the weather is sultry and often cloudy. The monsoon arrives first in the southernmost coast of the Andaman Sea (see Tavoy, Kampong Ulu, Mergui Islands), around May 10th to 15th, and is very intense from the very beginning, so much so that in this month, in this region more than 400 millimetres (16 inches) of rain fall. In Yangon, the monsoon arrives between May 20th and 25th, in Mandalay (where, however, it's less intense) between May 25th and June 1st, and finally it arrives in the far north between the first and 10th of June.
Here is the average precipitation in Mandalay.
Average precipitation - Mandalay
MandalayJanFebMarAprMayJunJulAugSepOctNovDecYear
MandalayJFMAMJJASONDY
Prec. (mm)4314014011585135150125406840
Prec.(in)0.20.101.65.54.53.35.35.94.91.60.233.1
Days00038769873153

The rainiest months are almost everywhere June, July and August, and the rains in the two coasts exposed to the west (the north coast on the Bay of Bengal and the south coast on the Andaman Sea) are plentiful, in fact in some areas even more than 1,200 mm (47 in) of rain per month fall! Yangon is a bit less rainy, because it's located in the area of the Irrawaddy river mouths, which are exposed to the south, and receives about 500/600 mm (20/23.5 in) of rain per month during this period, while as mentioned the central area of Mandalay receives just 100/150 mm (4/6 in) of rain per month.
Here is the average precipitation.
Average precipitation - Yangon
YangonJanFebMarAprMayJunJulAugSepOctNovDecYear
YangonJFMAMJJASONDY
Prec. (mm)527153005455606003702056072681
Prec.(in)0.20.10.30.611.821.52223.614.68.12.40.3105.6
Days000213252626201250129

In September, rainfall declines a bit, except in the extreme south, while during October the monsoon withdraws, starting from where it came last, therefore from the north; the monsoon is replaced by the drier, and gradually cooler winds, which begin to blow from the north or northeast.
The monsoon affects particularly the coast of the Bay of Bengal and the Andaman Sea, being exposed to the southwest winds. So this is the wettest area of Burma, and the rainfall amount reaches 5,000 mm (195 in) in a year (or rather, in the rainy season, because in the rest of the year the rains are very little).
Here is the average precipitation in Sittwe (formerly Akyab).
Average precipitation - Sittwe
SittweJanFebMarAprMayJunJulAugSepOctNovDecYear
SittweJFMAMJJASONDY
Prec. (mm)1010545270109011551025535290105154555
Prec.(in)0.40.40.21.810.642.945.540.421.111.44.10.6179.3
Days000313232525181040121

By contrast, in the central plains, precipitation descends below 900 mm (35.5 in) per year, as it happens in Mandalay, Myngyan, Magway, and Bagan, the ancient capital of several Burmese kingdoms. On the contrary, at the foot of the mountains and on the slopes, the monsoon rains are plentiful.

Bagan

During the monsoon period, Burma can be hit by tropical cyclones. The cyclones that form in the Indian Ocean and could hit the coasts of Burma directly from the south-west are more dangerous, although they generally follow a more northerly trajectory, directly affecting Bangladesh. The area most at risk is the coastal area, which includes the capital and the Irrawaddy Delta, as seen in May 2008 with the terrible cyclone Nargis. The most intense cyclones are generally formed either in the first phase of the monsoon (May, but sometimes also late April) or the last (September-October). The cyclones that form in the Pacific Ocean and come from the east are called typhoons, and before reaching Burma they lose most of the energy over the countries most directly exposed, mainly Vietnam, and are therefore less dangerous for Burma.

For swimming, the sea is warm all year round, as it reaches its minimum of 25 °C (77 °F) along the northern coast (see Sittwe) in January and February. In winter, the northernmost coast of the Bay of Bengal can sometimes be affected by cool air masses, which can lower a bit the temperature.
Sea temperature - Sittwe
SittweJanFebMarAprMayJunJulAugSepOctNovDec
SittweJFMAMJJASOND
Sea (°C)252527293030292929292826
Sea (°F)777781848686848484848279

In the southernmost coast of the Andaman Sea, the sea in the winter months is warmer, not going below 28 °C (82 °F).

When to go

The best time to visit Burma runs from December to mid-February: the weather is dry and sunny, it's hot in the capital and along the south-central coast, while it can get cool at night or even cold in inland areas. In the second half of February, and even more in March and April it's definitely hot, but along the coast you can take advantage of the breezes and you can still swim and sunbathe, given that the weather stays good.
Wanting to travel in the rainy season, for example in July and August, you will prefer the central plains (see Mandalay, Myngyan, Magway), because they are more sheltered from the monsoon, while the coasts are definitely inadvisable because they receive heavy rains.

What to pack

In winter: in the north (see Mandalay, Myitkyina) and in low-mountain cities (Taunggyi, Kengtung), spring/autumn clothes, light for the day, a jacket and a sweater for the evening, hat and scarf for cooler evenings; in the south and Yangon, light clothing, a light sweatshirt for the evening, a scarf for the breeze, possibly a light jacket or sweater for cooler evenings; in the Himalayas, mountain clothing, hiking boots.
In summer: lightweight clothing of natural fibres, light raincoat or umbrella, comfortable shoes; a light sweater for the evening at low-mountain altitudes; in the Himalayas, warm clothes, raincoat, hiking boots.
To enter pagodas, it is customary to remove shoes. For women, it is best to avoid shorts and miniskirts.