Normally, we wouldn't say "go to the botanical gardens", but Kirstenbosch is not to miss, honestly probably one of the best gardens in the world!
Lion's Head is my favorite hike in Cape Town, it is steep but pretty short. A Lion's Head sunset/full moon rise is an especially amazing experience.
Table Mountain is what makes Cape Town so special, it has so many great hiking trails of all difficulty, don't limit yourself to Plattenklip Gorge (the main trail on the face)!
Llandudno is perhaps Jordan's favorite beach in Cape Town (but there are so many beaches all offering different things), just south of the 12 Apostles it has amazing views, and incredible granite boulders to explore.
Muizenberg is the main surfing beach (and warmer water beach) in Cape Town (well for beginning surfers and longboarding), if you are interested in renting a board and wetsuit write to Ant, ant@theantsnest.com and tell him you are Jordan's friend. He will also be at the wedding!
Cape Point is essentially the "Tip of Africa"... but not the most Southern point. It is cool and worth a visit if you have time, but not essential. One thing that is cool is seeing the convergance of the Atlantic and Indian Ocean.
Penguins! Worth a visit, is a tourist trap, but... Penguins!
For wine tasting nearer to the city than Stellenbosch and the wine region. Is also super bourgie, so there are some tasty restaurants as well.
Kalk Bay and Fishhoek are worth a visit, just down the road on False Bay from Muizenberg its a nice little fishing town (check out the Octopus's Garden for a funky cafe on the water in an old train). Fishhoek has nice beach on the Indian Ocean side.
If you want to check out one museum in Cape Town, I would recommend District Six, which tells the story of what happened to one of the only mixed districts of the city under Apartheid.
If you want to get your braai (South African BBQ) on township style, head to Gugulethu for Mzoli's. However, take a guide or go with a local, as it is in the township.
Personally, I think that Devils Peak has the best beer in Cape Town, their taproom in Woodstock/Salt River also has some tasty bites.
Basically Cape Town's central park, filled with gardens, museums... homeless people (or burgies as they are called in Cape Town), exactly like Central Park.
Big mall by the harbour, its got an aquarium as well!