Fanastic place for a holiday. I was there a couple of years ago, travelling around the Cape, Kruger and Namibia. I've been to Africa on several occasions and found South Africa (and Namibia), very accessible and safe, with the obvious exception of some areas around Joburg and Cape Town. We stayed in both Hout Bay and Cape Town itself - Hout Bay felt a lot safer and pretty relaxing. But for true relaxation, head further along the coast or inland.
We didn't do the "Garden Route" which is the typical tourist route around the coast, didn't have the time and wanted to see more wildlife, the reason we spent most of the time in Namibia and Kruger. The tourist infrastructure is very sound, wasn't worried for our safety at all - where the might be slight risk, the local businesses usually provide security (guards with shotguns usually). Like travel in any country, don't make yourself an obvious target, SA is no worse than many western countries, although Joburg isn't somewhere I'd want to go on holiday to.
The Cape has plenty to see. Table Mountain, watching Whales from land at Hermanus, Bontebok National Park (very low key no big game but relaxing). Kruger is one of the best national parks in Africa - it has very good road and camp system, lots of people, which puts the "back to nature" safari snobs off a bit, but there are a lot of animals to see and you can escape the crowds, which are no worse than in some parts of Kenya. Similarly Etosha in Namibia is fantastic, black rhinos by the campsite etc. Next time, we'll go to KwaZulu which has some great parks too (and less crowded).
Another thing in its favour is the food, which is very good by African standards (let face it some African food isn't great, with the exception of fish!). Very cheap wine in the restaurant. Also flights are very easy to book, for not a lot of money, as low as £400 return. Plenty of choice too and most of the accomodation can be booked online.
Also don't worry too much about booking a "tour". It's a doddle to book a DIY trip from scratch, as easy as the US, even easier than Europe infact.
In summary South Africa is great introduction to Africa, a very easy place to take a holiday and once you've been you might be as hooked on Africa as I am!
10 day isn't long enough really, try 4 weeks instead. Go for it!

????