We stayed at Yiayia's house which is ideally placed to explore the old town including St Andrew's St, the Castle, Museum and a small zoo. The house was very well provided for a holiday and the hostess was extremely helpful, meeting us late at night and even providing a bottle of wine on arrival. We had all we needed. The area is tourist free and has a covered market for fruit and veg and many small shops in short walking distance for groceries etc.
Restaurants are plentiful but the Limassol Sailing Club which is about 4 miles east of Limassol is good value and friendly. The beach is within walking distance but if you want the nearest quiet beach to swim go to Ladies Mile which is out of Limassol on the Franklin Roosevelt and follow the signs. Other local beaches are very touristy so for quiet places drive West taking the B6 to the beaches past SBA Episkopi to Paramali, Evdimou and Melanda and further on to Paramali which is quite busy and developed. These have just one restaurant on the beach with good local food. The small restaurant leading to Curium beach beside the small monastry is excellent value.
There are numerous archeological sites. Go to Colossi Castle, Curium, Apollo's Temple, Amathus,Kouklia and for wonderful Mosaics don't miss Paphos. Here there is a good resaurant on the keyside with a live pelican to watch (not the one in the 60s!)
Car hire we did through Auto Union out of Paphos and was trouble free. Do not buy the extra insurance for WUG (wheels underbody glass) unless you're going off track.
We ate, what was for us, a normal breakfast , had a large lunch at restaurants and in the evening had cold meat salads with cheese and local wine and fruit ,all bought locally, which was proceeded by an apperitif sitting under the stars on the patio roof. What more could one ask for?