Most frequent flier programs place Uganda into the Africa zone. You can find some generic recommendations on how to get to East Africa here. I already have a blog with my recommendations for which programs to join if you are new to the world of miles and points.
The major gateway city for Uganda is Entebbe. Once you have been in the miles and points game for awhile, you will get a feel for which airline to use where but if you are just starting out Wikipedia will show you all the airlines that fly into Entebbe.
The only One World member serving Entebbe is Qatar Airways via Doha. You can reach Doha from anywhere in the world where Qatar flies to.
From North America you can use Egypt Air via Cairo, Ethiopian Airlines via Addis Ababa, South African Airways via Johannesburg or Turkish Airlines via Istanbul.
From Europe you can use Egypt Air via Cairo, Ethiopian Airlines via Addis Ababa, South African Airways via Johannesburg (a bit out of the way but UA might allow it) or Turkish Airlines via Istanbul.
From Australia and New Zealand there are no direct flights, you need to get to Singapore or Bangkok, then you can pick up Egypt Air, Ethiopian Airlines or Turkish Airlines.
From North America you can use KLM or Kenya Airways via Europe.
From Europe KLM flies direct from Amsterdam with connection to most European cities or Kenya Airways.
From Australia Skyteam isn’t the best option unless you travel via Dubai and pick up Kenya Airways from there.
Etihad and Emirates serve Entebbe from all over the world via their hubs in Abu Dhabi and Dubai respectively..
HOW MANY MILES WILL IT COST?
Every airline member of the 3 main alliances has it’s own frequent flier program. Examples are in my East Africa post.