Have a look at a flight search engine (google flights, or kayak or skyscanner and select 'or nearby airports') and you'll see that it is not possible to get to and from a lot of your wishlist in a weekend. There are few places you could fly to on a Friday evening, given how long it takes to get to Entebbe from the centre of Kampala, so realistically you could spend some time in Kigali, Bujumbura, Addis Ababa or Nairobi. I agree with Johan that the draw to East Africa is it's natural environment. Of the African cities I have visited I would only spend time in Cape Town, the rest are functional and generally dirty, noisy with horrendous traffic problems. You can also get to Kilimanjaro (JRO) direct on Sat morning and leave on Sun evening, so you could see some savannah, forest and waterfalls. You should try and visit a safari destination as well, either in Uganda or via Nairobi and a charter flight, even if only for one evening and morning.
I only scratched the surface of Uganda and it was many years ago, but I loved the place and the people and I will go back at some point! Obviously there is gorilla trekking in Bwindi, or across in Rwanda or Congo, but I know it has become prohibitively expensive for most people. But I would highly, highly recommend a weekend at Kibale NP to see the chimpanzees. There are other monkeys and wildlife there too. Travel via Fort Portal is good (the road is sealed) and possible in a weekend from Kampala.