Map from Google - Uganda

Much of Uganda is occupied by a plateau, at an altitude of 1,000/1,200 metres (3,300/4,000 feet), where the climate is pleasantly warm, with average temperatures ranging between 21 °C and 23 °C (70 and 73 °F), and annual rainfall between 1,000 and 1,500 millimetres (40 and 60 inches). Temperature variations throughout the year are little, however, there is a warmer period from December to March, more noticeable in the north, and a cooler period from June to September. In general, temperatures are pleasant, although sometimes it can get hot during the day, especially from December to April, while nights can be cool or even cold depending on altitude, throughout the year.
In the northwest, around Lake Albert, whose banks are located at 620 metres (2,000 feet) above sea level, and along the Albert Nile River, the altitude drops below 1,000 metres (3,300 feet), and the heat becomes more intense. The western part of the Victoria Nile River, where the Murchison Falls are located, flows at an altitude lower than 700 metres (2,300 feet). On the contrary, in other areas the altitude exceeds 1,500 metres (5,000 feet), and at night it can get definitely cold.
The period from December to March is the warmest, because the prevailing currents come from the north-east, originally from the arid regions of Sudan and Eritrea. On the contrary, from June to September, the relatively cool south-eastern currents from the Indian Ocean and from the highlands of Tanzania prevail, at least in the southern part of the country, while the north is dominated by the more humid currents from DR Congo.

Rwenzori Mountains National Park

With regard to the rains, they occur in the form of downpour or thunderstorm in the afternoon or evening: even when they don't bring heavy rains, thunder and lightning storms can erupt throughout the year, especially in the south. Typically the least rainy period for the whole country corresponds to the months of January to February, which is an anomaly in comparison with tropical areas, where the dry season is usually the least warm. In the rest of the year, the rains are distributed in different ways depending on area: the south, crossed by the Equator, has two rainy seasons, from March to May (called long rains) and from late September to December (called short rains), alternating with two relatively dry periods (January-February and June-August). This happens because at the Equator the sun passes overhead twice a year (on late March and late September), and for the thermal inertia of the ground and the atmosphere, the effect continues in the following weeks.
For instance, here is the average precipitation of Kampala, where precipitation amounts to 1,260 mm (50 in) per year.
Average precipitation - Kampala
Prec. (mm)7065130170120706595110140150901265

On the other hand, in the centre-north there is a single rainy season from April to October, and a dry season from December to February.
Here for example, the average rainfall in Gulu, located in the north, at 1,100 metres (3,600 feet) above sea level.
Average precipitation - Gulu
Prec. (mm)10408017020014016022017016090401480

The driest areas are the south-west, bordering North Kivu and Rwanda, and the north-east: in both areas, precipitation drops below 1,000 mm (40 in) per year, but the north-east, near the arid areas of Kenya, is more subject to drought.
Here is the average precipitation in Moroto, located in the north-east, at 1,400 metres (4,600 feet) above sea level, where 810 mm (32 in) of rain per year fall.
Average precipitation - Moroto
Prec. (mm)1530551001057511511560605025810

The wettest areas are the centre-north (Gulu District), and some shores of Lake Victoria, where rainfall reaches 1,500 mm (60 in) per year. Lake Victoria, located in the south, at 1,100 metres (3,600 feet) above sea level, is large enough to affect the climate along its banks, reducing a bit the daily temperature range while increasing moisture. Another lake of some importance is Lake Edward, to the west, at 950 metres (3,100 ft) of altitude.

In the capital Kampala, which lies practically at the Equator, a short distance from Lake Victoria, daytime temperatures go from 29 °C (84 °F) in the period from January to March, to 27 °C (81 °F) from June to August; night temperatures are even more stable, hovering around 17/18 °C (63/64 °F) throughout the year.
Here are the average temperatures of Kampala.
Average temperatures - Kampala
Min (°C)181818181818171717171818
Max (°C)292929282727272728282728
Min (°F)646464646464636363636464
Max (°F)848484828181818182828182

In Kampala the amount of sunshine is decent, but not excellent, for most of the year, and decreases a bit in the long rains season, from March to May.
Sunshine - Kampala
Sun (hours)776667777777

In Uganda there are mountainous areas, with volcanoes and high mountains. The Ruwenzori Mountains are located in the west, near the border with the Democratic Republic of Congo, and reaches its highest point in Mount Stanley, which is also the third highest peak in Africa (after Kilimanjaro and Kenya), with its 5,109 metres (16,762 feet). In this and other peaks of the range, above 4,500 metres (14,800 ft), glaciers are found. Other high mountains are located in the east: Mount Elgon, 4,321 metres (14,176 feet) high, Mount Moroto, 3,083 metres (10,115 feet) high, and Mount Kadam, 3,063 metres (10,049 feet) high. In addition, in the south, on the border with Rwanda and Congo, we find the Virunga Mountains, whose highest peak (at least among those in Uganda) is Mount Muhabura, 4,127 metres (13,540 ft) high. Along the mountain slopes, cloud forests grow, where the air is mild or cool, moisture is high and sunshine is scarce. The Bwindi Impenetrable National Park is located in the south-west, between 1,500 and 2,500 metres (5,000 and 8,200 ft) above sea level, and is a UNESCO World Heritage Site, hosting the mountain gorilla, a species at risk of extinction.
Here are the average temperatures of Kabale, located near the park, at 1,900 metres (6,200 ft) above sea level.
Average temperatures - Kabale
Min (°C)11111112121091011111211
Max (°C)242424242324242425242324
Min (°F)525252545450485052525452
Max (°F)757575757375757577757375

As mentioned, the southwest is one of the driest areas of the country. In Kabale, about 1,000 mm (40 in) of rain per year fall, although the only really dry months are June and July. Here is the average rainfall.
Average precipitation - Kabale
Prec. (mm)6580115140100302055100110115901015

The Rwenzori Mountains National Park is significantly higher, since it is home to the aforementioned highest peaks of the range of the same name, and of all the country.

When to go

Since the temperature is quite stable and both the intense cold and the heat rarely occur, the best times to visit Uganda are related to the rains. January and February receive little rainfall almost everywhere, therefore they are the best months, although some showers may still occur, especially in the south and along the western mountain slopes, while during the day it can get hot especially in the north and below a thousand metres (3,300 feet). It is worth while to recall that the equatorial sun's rays require appropriate protection, especially in the mountains.
Even June, July and August experience little rainfall, but only in the south, therefore, they are fine if you intend to visit Kampala, Lake Victoria and southern parks such as that of the Ruwenzori.
The rainiest months in the south are April, May, October and November, while in the north, as mentioned, there is a single rainy season from April to October.

What to pack

All year long, for Lake Albert and at lower elevations, light clothing, a sweatshirt and a light jacket for the evening; for Kampala and Lake Victoria, light clothes for the day, sweater and jacket for the evening, light scarf for the breeze, light raincoat or umbrella. For the plateau, at higher altitudes, spring/autumn clothes, sweater and warm jacket, raincoat or umbrella; for the mountain peaks, warm clothes, down jacket, hat, gloves.