Limassol has a genuinely good restaurant scene for a city of its size — the meze tradition means you can eat very well without breaking the bank. Traditional meze restaurants serve a procession of small dishes: dips, grilled halloumi, loukanika sausages, fresh salads, and whatever the kitchen considers special that day. Order meze and let it arrive at the kitchen's pace.
The best traditional restaurants are concentrated in the old town and around Germasogia. The waterfront east of the city (toward Amathous) has a strip of smarter fish restaurants. Avoid the obvious tourist restaurants on the main promenade — the better places are found a block or two back. Meals in Cyprus are eaten late: lunch from 1pm, dinner rarely before 8:30pm.