You will see people running and walking everywhere, particularly on weekends. An extremely popular exercise spot nearby are the Westcliff steps, a flight of stairs that leads from Westcliff Drive up to the ridge. On a Saturday morning, you might have to stand in line to start your walk or run up. The park around the Zoo lake is beautiful for running and walking, too. You can also use the public outdoor pool there.
The closest gym is Milpark Virgin Active, which is literally a four-minute drive (unless during rush hour from 06.00-08.30) away at 39 Stanley Road in Auckland Park. Entrance is through the parking garage opposite the Garden Court Hotel. You can get a day or week access, depending on your needs. It has a reasonable indoor pool, too, plus sauna and steam room.
Mon - Thursday: 05.00-21.00
Sundays/public holiday: 06.00-14.00
You can play tennis at Parkview Tennis Club which is an about 10 min drive away.
The nearest and well maintained golf course is Parkview Golf Club. Also very close is Killarney Country Club.
Cycling in the area is probably only recommendable on a Sunday morning when there is no traffic. If you are really into cycling, best option is the empty roads and cycling lanes at the Cradle of Humankind.
You will also find a few fitness equipment items in the cupboard (set of weights, jumping rope, weight ball).