May and June are the best months: the weather is warm but not brutal, the INDA Greek drama festival runs performances in the ancient theatre throughout both months, and the summer tourist hordes have not yet arrived. September and October are a close second — still warm enough for swimming, quieter, and with the advantage of lower prices.
July and August are hot (35°C or above is common), crowded, and expensive. Ortigia in August can feel overwhelmed by tourists; the archaeological park in midday heat is genuinely unpleasant. If you visit in summer, start early — be at the Neapolis by 8am and take a long lunch break.
Winter (November through March) is mild by northern European standards but can be wet and the days are short. The town is quiet and accommodation is cheap; the archaeological sites are less pleasant in rain. The cathedral and museums are unaffected by weather.