This will be amazing. Cape town is small but it's worth staying different places if you are there for so long. So maybe a week city centre/Atlantic side, week in winelands, week false bay side.
Previous suggestions good. To take further:
Sunday night concerts at kirstenbosch botanical gardens are a classic. Book a picnic and pick it up beforehand. Fabulous atmosphere, wine, food etc with great music. Sometimes big local bands, sometimes orchestra's etc. Quick Google will find options.
13 Apostles Spa is insanely expensive but amazing. The rooftop bar is also great. Treat yourselves to a treatment, some.time in spa and a drink (and dinner).
Hout bay is incredible. Do some quick googling because I can't remember which days but there are a couple of fantastic markets that are worth a visit. Plus you can't move without finding an amazing restaurant/pub with great sea food. Mariners wharf is my favourite.
You can also take a boat from.hout bay to see seals etc.
There's also a good market at kirstenbosch (outside gardens) on a Sunday.
If you want to hike table mountain get a guide. If you aren't local, it's better. If you don't want to, stick with well known "easy" routes like constantia nek. Climbing lions head late afternoon to watch the sunset is incredible. But I haven't done it in years and suspect a guide is essential.
The waterfront is popular for shopping and tourists but for more "normal" shopping try one of the suburban malls. Cavendish square in Claremont, or there are a loads the northern suburbs.
If you enjoy cricket, see if there are any games at Newlands. It's an incredible ground.
Kalk bay is great for lots of arty stuff and a more funky edge. Fancy amazing restaurants (do visit Harbourhouse) sit side by side with Kalky's which is basically a fish and chips shop but where the fish is caught by the owner manager and the variety of options and quality will blow your mind!
Not great with specifics but definitely spend time in the winelands. Stay at one of the farms (or more than one), where accommodation, food, scenery and wine will all be amazing.
If you want to try shark diving (cave), there are options slightly up cost.