Lived in Constantia for a year - great place. If you're sensible then its no riskier than any other big city.
Top stuff to do?
The townships just to get a sense of the place - we drove through a few (by mistake) and it was a little uncomfortable but you can get legit guided tours from the big hotels which are well worth doing.
Waterfront for the nightlife.
Bolders beach for the penguins.
Franshoek / wine route is great for a couple of days.
Paarl if you like golf (maybe not for a honeymoon though) and top red wine.
Table mountain, obviously.
Rhodes memorial - nice afternoon stroll.
Lions Head for a great view back across the bay.
Surfing at Muisenburg
Garden route is great. Knysna and Plett both very nice and if its the right time of year the whales at mossel / false bay. Titsikama (sp) national park on the way up to PE and the wilderness both stunning - plenty of bungee opportunities along the way if you get the urge.
Didnt like PE - would be tempted to fly up from there to mpumalanga then spend 4 or 5 days self drive around kruger staying the the national park axxomodation rather than one of the lodges outside. If thats your last stop you can either fly or drive back to joburg for the flight home.