Best method is to fly from Joburg to KMIA (Flights every hour and only 40 Minutes), and then a Private Charter from KMIA to Garonga - only a 30 Minute flight. The Garonga Airstrip is 5 minutes from the main Camp.
Direct flights are also available from KMIA - CT.
Hoedspruit and Phalaborwa are the 2 closest airports, about 75 minutes drive away from the camp.
Transfers from the airport to Garonga and back can be booked with reservations or you can make your own road transfer booking with local transfer companies.
The advantage of Phalaborwa is that there is a late afternoon flight to Johannesburg during the week and on Sundays in which you can connect with international flights back to UK, Europe, etc.
There are 3 x flights per day except Saturday where there is only one.
The advantage of Hoedspruit is that there are 3 x flights per day around mid-day throughout the week and direct Flights from Cape Town to Hoedspruit and vice versa on Tuesdays, Thursdays and Sundays.
A flight to KMIA from Johannesburg and then a road transfer to Garonga is a waste of time. you may as well drive from Johannesburg to Garonga direct. The times will be the same anyway. Likewise return to Johannesburg.
Another Way: Air Charter from from/to Hoedspruit or Phalaborwa from/to Garonga. 20 Minutes flight door to door.