Map from Google - Uganda

The majority of Uganda is occupied by a plateau, at a height of 3,300/4,000 feet, where the climate is pleasantly warm, with average temperatures ranging between 70 °F and 73.5 °F, and annual rainfall between 40 and 60 inches. The 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, the temperatures are pleasant, although sometimes it can get hot during the day, especially from December to April, while at night it can get cool or even cold depending on altitude, throughout the year.
In the northwest, around Lake Albert, whose banks are located at 2,000 feet and along the Albert Nile River, the altitude drops below 3,300 feet, and the heat becomes more intense. Even the western part of the Victoria Nile River, where the Murchison Falls are located, flows at an altitude lower than 2,300 feet. On the contrary, in other areas the altitude exceeds 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 is from January to February, which is an anomaly compared 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 50 inches per year.
Average precipitation - Kampala
Prec. (in)

In the center-north instead 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 3,600 feet above sea level.
Average precipitation - Gulu
Prec. (in)

The driest areas are the south-west, bordering North Kivu and Rwanda, and the north-east: precipitation drops in both areas below 40 inches per year, but the north-east, near the arid areas of Kenya, is more subject to the risk of drought.
Here is the average precipitation in Moroto, located in the north-east, at 4,600 feet above sea level, where 31.8 inches per year fall.
Average precipitation - Moroto
Prec. (in)

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

In the capital Kampala, which lies practically at the Equator, daytime temperatures go from 85 °F in the period from January to March, to 80 °F in the period from June to August; the night temperatures are even more stable, hovering around 63/65 °F throughout the year.
Here are the average temperatures of Kampala.
Average temperatures - Kampala
Min (°F)646464646464636363636464
Max (°F)848484828181818182828182

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 16,762 feet. In this and other peaks of the range, above 14,800 feet, glaciers are found. Other high mountains are located in the east: Mount Elgon, 14,177 feet high, Moroto, 10,115 feet high, and Kadam, 10,049 feet high. In addition, in the south, on the border with Rwanda, we find the Virunga Mountains, whose highest peak is Mount Muhabura, 13,000 feet high. Along the mountain slopes, cloud forests grow, where the air is mild or cool, moisture is high and the sunshine is scarce. The Bwindi Impenetrable National Park is located in the south-west, between 5,000 and 8,200 feet 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 6,200 feet above sea level.
Average temperatures - Kabale
Min (°F)525252545450485052525452
Max (°F)757575757375757577757375

As mentioned, the southwest is one of the driest areas of the country. In Kabale, about 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. (in)

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 determined by the rains. January and February have little rainfall almost everywhere, and 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 3,300 feet. Recall that the equatorial sun requires appropriate protection, especially in the mountains.
June, July and August experience little rains, but only in the south, therefore, they are fine if you intend to visit Kampala, Lake Victoria and southern parks such as the Ruwenzori.
The rainiest months in the south are April, May, October and November, while in the north 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.