Hi OP, I have been to SA many times, you will love it. Your choice of Cape Town and Kruger is perfect, in fact I am doing the same later this year.
I have never visited Joberg and although I understand it has some interesting attractions I wouldn't include it for a first time trip. Same advice goes for visiting KZN (Kwa Zulu Natal and not Natal as it was referred to previously) Although it has everything, beaches, wildlife mountains and history save it for another time.
Your choice of safari destination is firstly dependent on whether you are travelling with children as a lot of game drives don't allow children under a certain age. Also no-one has mentioned yet malaria, If you want to go to Kruger (the national park or the private reserves bordering it) you need to take Malaria prophylactics.
If you do choose Kruger, you can fly directly into the park (Skukuza) from Cape Town. From here you either pick up your rental car and start game driving as soon as you leave the airport, or if you have decided on a private lodge they will collect you. You asked about game drives in Kruger. To book these in advance, which I suggest, you need to have first of all booked the accommodation, this is because the game drives leave and/or return whilst the gates to the rest camps are closed. To book and use the SANparks website there is a brilliant guide on the South Africa TripAdvisor forum.
As to how long you spend in a park I can't say, this time we are going for 7 nights and staying in 4 different rest camps. If you are not sure if you will take to it maybe 4 nights in 2 different camps, or a combination of a private lodge and a couple of days self driving.
Good luck with your itinerary planning.