A great way to see around - very cheaply - is on the
red bus. You can get on and off where you like, there is another bus every 20 minutes or so, and it passes by all the popular tourist attractions.
Some of my own favourites are Table Mountain, the Aquarium, District Six museum, Boulders Beach (hold your nose though, those penguins stink!), the V&A Waterfront and Afternoon Tea at the Mount Nelson. Another fascinating tour to take is a township tour, I think it was the highlight of my stay in Cape Town.
For a quirky museum visit, it's hard to beat the one at Groote Schuur hospital where Dr Barnard carried out the first heart transplant in 1967. The rooms that the transplant was carried out in are set up in an authentic representation of the operations.
If you hire a car, it's well worth driving around the peninsula and through the National Park. The baboons are very cheeky and you can get great photographs. A warning from a friend - if you are in a convertible car, don't sit there eating anything, the baboons will take it off you! Make sure the passenger in the car has a camera handy when you're driving along Chapmans Peak Drive, between Noordhoek and Hout Bay, as the scenery is utterly spectacular.
Fabulous place and fabulous people. However, do heed the warnings about your personal security and be extra careful in the evenings.