Given the vastness and scope (not to mention the widely varying types of travel) of Africa, we can't comprehensively cover this topic in one shot on this page. It's like asking the question: How do I get to South America?
Once you've decided which part you're going to, then we can start talking about international hubs.
For Southern Africa, the main hub is Johannesburg (or Jo'burg), South Africa. From there, you're likely to take connecting flights to other places around Southern Africa (e.g. Livingstone (Zambia), Lusaka (Zambia), Victoria Falls (Zimbabwe), Harare (Zimbabwe), Durban (South Africa), Cape Town (South Africa), etc.). We flew nearly 14 hours from Sydney to Johannesburg with Qantas (though I think they're code-shared with South African Airways or SAA). I think British Airways also has flights from London to Johannesburg. From Jo'burg, we flew about 90 minutes to Livingstone, Zambia.
For East Africa, the main hub is Nairobi, Kenya. Kenya Airways serves much of the regional flights here. In a few cases, there may be some carriers flying direct to Arusha, Tanzania or even Dar Es Salaam, Tanzania. In our experience, we took a flight (with a connection) from Lusaka, Zambia to Nairobi, Kenya [total flight time including connection was around 4.5 hours]. Once in Nairobi, we took local flights to Arusha (Tanzania) [about 1 hour] and Entebbe (Uganda) [about 1 hour].
For Northern Africa, I'm not sure what the hub is, but if I had to guess, it's probably Cairo, Egypt. We flew there with a connection in Khartoum, Sudan from Nairobi [about 4.5 hours total with the connection] via Kenya Airways.
Once we got to our destinations, we were pretty much on escorted tours (Victoria Falls was an exception, but we did have someone meet us there). Therefore, there was a representative meeting us at each airport. If you're on your own, I can't offer any additional advice on the matter since we didn't go it alone (except for Victoria Falls).
Speaking of which, I should mention that if you're spending a considerable amount of time on the road to get from place to place (which is very likely), then you're going to have to allocate plenty of time to get from place to place. If you're on an escorted itinerary, this shouldn't be much of a problem, but you should still double check as tour operators sometimes makes mistakes.
So why do you need plenty of time?
Most African roads are littered with potholes if they're on tarmac or they're full of ruts and washboards if they're unsealed. Many taxis and safari vehicles are second hand vehicles (usually Toyota vans or Nissan Pajero [or less common, the robust Toyota Land Cruisers]). Flat tires are common and breakdowns are not out of the question. Even getting stuck in the mud or bridges/roads getting washed out can happen. The bottom line is don't try to push the tour operators or your guide into overly ambitious itineraries (because that's just asking for failure; trust us, we've experienced it)!
This is pretty much all I have to say about the subject of getting to and around Africa. Like I said earlier, I'm sure there's much more to say about it, but it would be beyond the scope of our experiences and hence we'd have to defer such comments.
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