First decision is choosing a place to stay. To begin with, I recommend having a car in any case. Distances are definitely not walkable if you want to see even a fraction of all what is out there. :)
Choosing the best area depends on what you are looking for, so my tips are just a brief overview and based on my preferences. A surfer will probably value beach location more than night life and so on. I personally value a lot that the sea is close, and that I'm on a peaceful and beautiful area yet close to the services like restaurants, yoga studios etc. If you’re in Cape Town for a longer time, I’d stay in couple of different locations to experience more.
Regarding safety, I personally find all of the areas listed below safe.
Here's a list of areas I'd primary look to stay in:
Anywhere between BAKOVEN and DE WATERKANT
Quite a wide range but I’d happily stay anywhere here, including Camps Bay, Bantry Bay, Clifton, Fresnaye, Sea Point and Green Point.
Each one is by the beach with beautiful coastal line and amazing views. There are several beautiful beaches here, public and more hidden ones;). I’d prefer Bakoven side because of its beauty, peace and stunning beaches. In Sea Point and Green Point, I’d choose up the hill instead of close to promenade, for its views and nicer streets. De Waterkant is a small area with great location and beautiful streets.
Gardens is full of nice (but not fancy) restaurants, bars and affordable accommodation. Close to centre and beach, walking distance from centre (I would avoid staying in the very centre).
Other nice areas:
Tamboerskloof is lovely area, I would happily live there or at least go check it out once, the wealthy locals live here, if you go deep enough...
Woodstock is a more “rough”= hipster area with a lot of art galleries, restaurants, weekend markets etc. I like it a lot but personally prefer being closer to the coast line and mountains.
NEWLANDS, CONSTANTIA, RONDEBOSCH
Extremely beautiful areas where many wealthy locals live. Close to the botanical garden, very peaceful. Minus sides that its not by the beach and there are not many services, it’s mostly residential.
HOUT BAY & KOMMETJIE:
Beautiful areas by the beach. Quite far down from the centre (approx 35-45min drive without traffic) , but great for peace and beautiful views. If I'd only want to surf or do yoga I'd happily stay here for few days, but after would start missing the buzz closer to centre.
KALK BAY, MUIZENBERG:
Here I feel like being in a fish village somewhere in Southern Europe. Great vibes but same as above, not enough of buzz for me personally. :)