I am originally South Africa and have just got back from 3 weeks in Cape Town 
Joburg is definitely a viable airport if you want to do a safari. It is nearest the big game parks and there are loads of low cost flights from there to Cape Town.
Like Chicken says it is worth having a car in Cape Town and there is loads to do. My DC are younger but their highlights were going up table mountain on the revolving cable car, the aquarium and a cruise on a pirate ship. There is also an amazing new walkway on top of the trees in the botanical garden
boomslang
But you may also want to consider Durban for beaches, because the water is warmer and then the drive from Natal to Cape Town down the garden route is beautiful.
As for season - Easter is autumn there, but it will still be fairly warm. If you mean early summer there, that will be Oct/Nov and a very nice time to go, but if you mean early UK summer, it will be winter but still fine - cape town has winter rainfall, but we only had a couple of days rain in 3 weeks and the temp was in the 20s most of the time. Joburg and inland will be cooler though, especially at night.
Let us know what you decide to book - I'm sure you'll have a lovely time wherever and whenever you go 