I've hunted plains game in Africa a couple of times. Once in South Africa in 2008 & in Zimbabwe last year.
Hunting in Africa is highly regulated & you will have to use a guide.
In South Africa most of the hunting is done on privately owned concessions or game farms. In Zimbabwe there is a mix of privately owned concessions & government parks to hunt in. Either way there is basically a daily rate, which covers accommodation, guides, vehicles, meals, drinks, and a separate trophy fee per animal. Every animal taken is retrieved for skinning & the meat and innards are always used.
How you conduct your hunt is up to you, but I've always talked to the guide prior to the hunt to confirm that I prefer stalking the bush, rather than shooting from a vehicle. All hunts I've been on have been fair chase, waking before sunrise & walking upwards of 10 miles per day. The hunts I've been on have not guaranteed success, as I've still to bag a Kudu, but you will not be disappointed in the numbers of game seen.
Costs vary considerably depending on the level of accommodation & the type of game hunted. A 10 day plains game hunt package including half a dozen trophy fees can be had for around $10,000. On top of this are your airfares & any side trips. I would also recommend you allow a day or so either side of your hunt dates, as things in Africa don't always happen on time.
If you looking at hunting dangerous game get a quote, but look at around $15,000 for cape buffalo , $15,000-$20,000 for Leopard & a lot more for Lion.
Each time I've been to Africa I've spent a month- 2 weeks hunting & 2 weeks travelling around. Each time I've been I have had a fantastic trip & have always said I'd return.