There's Kruger National Park. For Kruger you need time and planning ahead.
I didn't have time nor did I plan ahead.
Near the end of my trip to South Africa I decided, that one cannot go to Africa without seeing big game and those gumtrees. You know, see that cliché image of Africa: Setting sun, big tree, big game.
Close to Cape Town are a couple of Safari lodges. Some of them dodgy, some of better reputation. Via couchsurfer.com I found a fellow traveller to join me for this spontaneous trip. The lodge I chose wasn't the most expensive, because it was young.
The reserve wasn't yet self-sustaining, so they had to feed the animals (or so they said - maybe it was also to generate better photo ops).
The feeding ground is where most pictures were taken.
At night, armed guards protect the rhinos. Even this lodge had lost animals due to poaching.