Welcome to the climbing community Peet! You and your wife have certainly made a great decision to take up rock climbing as a hobby!
Gear that's required for indoor climbing (at climbing gyms such as CityRock or climbingbarn.co.za):
1. Shoes
2. Harness
3. Chalk & chalk bag (if your hands sweat easily)
When climbing outdoors, you'll additionally need:
4. Rope & belay device (also needed for indoor lead climbing)
5. Quickdraws (between 10 and 14 should get you up most routes)
6. 2x slings with screw gates
Advisable:
7. Helmets for the safety of both the climber and the belayer
8. Rope bag with rope tarp to keep your rope clean and for better rope management
9. Route guide so that you know where to find the routes, their grade, height, and number of quickdraws required. Route information is also available online at thecrag.com and at climbing.co.za/wiki
There are a couple of things to learn and the staff at climbing gyms should all be able to teach you the knots and how to belay properly (toprope & lead). At CityRock you can get toprope belay tags and lead belay tags after passing their belay tests. These belay tests must be done on two separate days and are free of charge, but you'd still need to pay for a day pass if you're not a CityRock member.
Most climbing gyms have all the necessary climbing gear for sale. On the Garage Sale forum you can find second hand and new gear looking for new owners: climbing.co.za/forum/viewforum.php?f=18 also: vertigogear.co.za/collections/2nd-ascent
More than 99% of the climbers I've met are very friendly and sharing and happy to teach you the knots, how to belay safely, how to lead climb and how to clean the route, i.e. getting all your gear off the route once you've reached the top. And most of them are also willing to let you climb up routes using their quickdraws and rope in the beginning until you've bought your own gear. As long as you never step on their rope!!!

You'll learn to understand this rule very well as you spend more time at the crag.
Arguably the country's best sport climbing destination with over 1000 sport climbing routes is Waterval Boven in Mpumalanga, about 2½ hours from Pretoria. I'd recommend you go spend some weekends in Boven, even as a complete noob, since there's an abundance of beginner-friendly routes.
Enjoy the journey!