We did a trip from Kampala, Uganda, to see chimpanzees and gorillas, then back through Kampala and Nairobi (by bus) to Arusha, Tanzania. 4 day camping safari and then back to Nairobi to fly home. Took about 17-18 days and cost about £1,400 each 10 years ago. Probably my favourite holiday of all time, even though we got salmonella! I booked the permits, safari and planned for transport before we went because we had limited time.
The cost of the gorilla treking permits, safari and fuel (i.e. transport) will have gone up a lot since then. But if you are interested in wildlife I would recommend gorilla trekking (the Rwandan side would be just as good) and try and see chimpanzees too. We went to Kibale near Fort Portal which I would highly recommend for a 2-3 day stay. You could alternatively do Rwanda, Burundi and then Gombe in Tanzania (where Jane Goodall did most of her research). I have looked at it, but would personally want to take 3 weeks to do it properly. Uganda was incredibly friendly and back then the tourism industry was very limited.
My partner went to Malawi as part of an overland tour about 15 years ago and loved that too; very friendly and easy going part of the world.
To reduce costs, and increase the likelihood of great anecdotes for when you get home, use public transport. But expect it to breakdown or just not turn up. Which is why you need time! A lot of Uganda / Rwanda / northern Tanzania and Malawi away from the main highway are gravel roads. Is the £2k including flights? As above, Lonely Planet and Rough Guides to East Africa are a good start.