The poor reputation of the Central Business District (CBD) is no longer deserved. There are numerous galleries and art spaces popping up across the CBD. The city centre is the most-visited part of Johannesburg for African tourists, particularly the traders who come to shop at Johannesburg's wholesale outlets.
Newtown and the Market Theatre area (the city's cultural precinct) is now easily accessible from the highway and Mandela Bridge and very fun; here you will find live music venues and bars too.
Braamfontein, the university area, has a great Saturday market, lively night life, is very artsy. By day there's nothing here.
On the east side of the city, Main Street Life, Maboneng Precinct and Arts on Main (especially the Sunday market and The Bioscope independent cinema). The Maboneng area has 24-hr security so you can freely walk around the streets, just be sure not to stray away too far. Troyeville has a fantastic restaurant at the Troyeville Hotel, an art centre and all the main sport stadiums (football, tennis, athletics, rugby).
On the west side, Fordsburg is the formerly-Indian now known as “little Somilia” and “ Eithiopia Town” part of central Joburg and has some Indian and Pakistani restaurants as well as African street food, shops and markets. Good food is to be found in this neighbourhood, which, by Johannesburg's standards, shows signs of street life in the evenings, and more so on Friday and Saturday. Most places are halal, and so no alcohol is served. The Oriental Plaza shopping mall is here and has good bargains.
In the centre of town, between Jeppe St and Bree Street at Delvers Street, look up and see the Amharic script which denotes that you are in the Ethiopian/Somali part of town. There are Ethiopian restaurants and coffee shops in the Africa Mall and Johannesburg Mall. Best to arrive before 14:00.
To the north east, Yeoville is known as Le Petit Kinshasa and is home to many of the Francophone African diaspora in Johannesburg. Lots of Camerounian restaurants and Congolese bars.
Hillbrow has a bad reputation for drug dealing, sex bars, crime, etc., but is slowly improving. If you visit Constitution Hill, or Johannesburg Art Gallery, Hillbrow is right across the street, not that scary. Empty your pockets and go for a walk to the base of the Hillbrow Tower on a Sunday morning. Stay on the main streets and keep your wits about you, and don't carry anything that is worth stealing. It is certainly an interesting experience. Worth watching Louis Theroux's Law and Disorder in Johannesburg before your visit.