Johannesburg sits at the centre of the Gauteng province and makes a reasonable base for day trips in several directions.
Pretoria (60km north, 45 minutes) is the government capital and has a different character from Johannesburg — quieter, more orderly, and with substantial colonial and Boer War-era architecture. The Union Buildings (Herbert Baker, 1913), the Voortrekker Monument, and the National Zoological Gardens are all worth seeing. The Gautrain connects Johannesburg to Pretoria in about 40 minutes.
Pilanesberg National Park (2 hours northwest) is the most accessible Big Five safari from Johannesburg and is malaria-free, which makes it suitable year-round. The park sits in an ancient volcanic crater and has good populations of elephant, rhino, lion, leopard, and buffalo. Driving yourself is straightforward with a 4x4 or high-clearance vehicle; guided day tours from Johannesburg are also widely available. Sun City resort is directly adjacent to the park if you want an overnight stay.
The Cradle of Humankind (50km northwest) is the UNESCO World Heritage Site where some of the oldest hominid fossils in the world have been found — including Mrs. Ples and Little Foot. The Maropeng visitor centre explains the science well, and the Sterkfontein Caves tour is excellent.
Cullinan (45 minutes east) is the town where the world's largest gem-quality diamond was found in 1905. The Cullinan Diamond, 3,106 carats, was cut and set in the British Crown Jewels. Underground tours of the working mine are available. The town retains its Edwardian mining character.