As a St. Louis boy who's been living in Cape Town for the past 7 years, I've had a lot of flights back and forth.
I often prefer the shortest flight options, even if that means having to fly economy. There are only 2 direct flights from the US to SA: Delta from ATL-JNB or South African Airways from JFK-JNB. Both are decent options with good economy space and amenities. And really nice to avoid having to go through Europe or the Middle East. Flight times are long (15-17 hours), but overall travel time is the quickest. Unfortunately, there are no direct flights from the US to Cape Town (although rumors about a Miami route have been floating around recently).
In terms of other options, I often fly Turkish Airlines. Can fly direct to IST from many US cities (ORD included) and then it's a direct flight from IST to CPT. Likewise, on the return, there would be direct flights from JNB to IST. Many times I've seen them offering upgrades to Biz Class at the airport for around $1000 per segment. I've used miles to upgrade to biz on Turkish many times and the product is top notch!
Have never tried PE, but other posters seem to have you covered on those options. My guess would be the shortest options through Europe are going to either be on British or KLM, which have direct flights both into CPT and out of JNB.