Check out http://www.newtown.co.za/news.php?id=216 - Newtown is a great place to visit with lots to do.
Agreed on many of the posts above - stay away from Hillbrow/Yoeville/Berea, but Melville, Maboneng, CBD, Parktown etc and the other suggesstions above will give you great insights. I often have to go into CBD for meetings or my dentist, and have felt safe. Lots of police presence, as well as cameras. It is just busy and active, and lots of people.
There are also the Randlord tours, and gardens of JHB, and the ghost tour - I have not been on these myself, but may be interesting for you to look into. I came across this, which may be useful to you urbanadventures.com/johannesburg_tour_Jozi_b…
As with any large city anywhere in the world, there are safe and unsafe places. Just be aware, don't act or look like a tourist, keep your valuable safe somewhere, try to find reputable guides and companies. South AFricans are generally friendly and helpful. If you have to draw money, rather do it before you go on a walk, or use your credit card, and only draw money inside a shopping centre, not on a busy street.
Otherwise, have FUN, and enjoy the hustle and bustle, sights and sounds and smells of our wonderful country.