I was fortunate enough to go to South Africa last year for a plains game hunt.
The big 5 animals you mentioned (rhino, cape buffalo, and elephant) require a 375 H&H as the minimum caliber allowed.
For plains game (someone mentioned impala, warthog, and a few others) a .30-06 or 270 win work just great.
Not sure about the lion but a friend of mine has taken leopards with a 270 win. The main thing is to use good bullets that will expand and stay together.
The most important thing is to use a caliber you can shoot accurately. My guide told a story about a client who brought his 300 weatherby magnum for plains game hunting. After the client missed several animals, the guide gave him a .30-06 to use and the client then successfully harvested several animals.
