If you're staying in Cape Town and want to see the great white shark in its natural environment, a short and beautiful drive to Gansbaai is a must do. Renting a vehicle from the Cape Town airport is easy, and for an economy size vehicle, it can be quite cheap. Just be prepared for a stick-shift on the left side of the road.
There are plenty of Highlights written about Cape Town and places to stay, but I suggest looking into Colona Castle in Muizenberg. A boutique hotel with a beautiful view and unique rooms, the breakfast is simply damned good, and the service great. Anyway, leaving Cape Town, you will hug the rocky coast heading east. The drive is of course stunning, especially in the morning with the sun just rising. There are some small towns you pass along the way to Gansbaai- I'd suggest stopping at Hermanus on your way back. Grab lunch at one of the outdoor restaurants where you can look out at the ocean and hope to see some whales.
In Gansbaai, Shark Diving Unlimited is easy to find just by using a map. They serve you breakfast then you get your brief on board. Once everyone has on their gear, you hop down into the cage and wait for the spotter to say "down." You have no snorkel and no SCUBA, so you hold your breath and go under, then watch as these enormous creatures swim around your cage, sometimes bumping into it. These animals are amazing, and the experience surreal. Bring an underwater camera, or use a polarizer filter on a dslr at the surface.