Gautrain is the rapid rail line connecting OR Tambo Airport, Sandton, Rosebank, Park Station (the CBD), Pretoria, and a few stations in between. For airport transfers it is by far the most reliable option: the train runs every 12 minutes at peak times, takes about 15 minutes to Sandton, and costs a fraction of a taxi. If you are staying in Sandton or Rosebank, the Gautrain is the obvious choice.
Uber is the practical answer for getting between neighbourhoods and for anywhere not served by the Gautrain. It is reliable, affordable by international standards, and the safest way to move around the city at any hour. Request your Uber from inside a building rather than on the street, and do not loiter while waiting. The app works exactly as you would expect.
Do not use minibus taxis as a visitor. They are cheap and they are everywhere, but the route system (conducted through hand signals and spoken shorthand) is opaque to outsiders, and conditions vary. For locals they are the backbone of the city's transport; for a visitor with a phone and an Uber account, they offer no advantage.
Walking between neighbourhoods is not advisable. Even the northern suburbs have poor pedestrian infrastructure and the distances between points of interest are car-scale. Within a precinct — the Rosebank strip, 7th Street in Melville, the Maboneng block — walking is fine during the day.
Driving is the other option if you are spending more than a few days and want flexibility. South Africans drive on the left. Traffic is heavy during rush hours on the N1 and M1 freeways, and parking in the northern suburbs is generally available at malls. Do not leave valuables visible in a parked car.