I would suggest Camps Bay - you have the mountains on one side, a gorgeous beach, it's a posh area with some expensive boutique hotels, but we stayed here and price per person was pretty reasonable: www.aubergeducap.co.za/index.php/units/ocean-suite (About £45 per person including breakfast, but depends on exchange rate obviously). Great views, breakfast on the patio when weather is good, only downside was the incredibly steep driveway! It's not near the beach if you walk but only 5 to 10 minutes (and maybe £2 or so fare) in an Uber.
Also some very nice places on airbnb which are even cheaper - can look up and link to the one we stayed at if you want (we stayed 2 nights at Auberge, then went to Franschoek and Hermanus for 3 nights each (airbnb) then came back to Cape Town for another 2 nights in an airbnb).
Some decent places to eat too - Codfather for seafood is quite good, Mantra Cafe was great - does breakfast, lunch (grilled calamari was delicious!) and dinner or just drinks, coffee was good too and has a view of the beach. www.mantracafe.co.za/menu/ Some people raved about the Hussar steakhouse, but we thought it was very average (we had great steak at Char in Hermanus though).
The hop on hop off tourist bus stops at the beach and you can use that to get to the Waterfront, CBD, or go the whole circuit which takes in some of the vineyards, Table Mountain, Hout Bay etc. Normally not a big fan of these tourist buses, but the Cape Town one is worthwhile and the commentary on the headphones is actually quite interesting.
Uber also works very in Cape Town, inexpensive and from Camps Bay to Table Mountain cable car station is only a short journey.
If you like wine, a day trip to Franschoek and a trip on the wine tram (book in advance) is recommended.
You could stay in the Waterfront area too - but it's a bit 'touristy' and shopping orientated. Sea Point looked okay too. I would avoid staying in the city centre (CBD) - on a previous visit I was constantly harassed and followed by aggressive beggars there and one even threatened me with a knife. :( A pity as there are some great places to eat and drink there - but better to stay elsewhere and get a taxi/uber to restaurants/bars there imo.
Also be aware of the severe water restrictions they have currently - no baths, take short showers etc.