The short answer
Late May to late June and September to early October are the sweet spots: warm sea, everything open, far fewer people than midsummer and a gentler wind. July and August deliver the full scene — and the highest prices, the biggest crowds and the strongest meltemi. From November to April the island is quiet and most beach venues are shut.
April and May — waking up
Orthodox Easter, whenever it falls, is the informal start of the season: hotels and restaurants reopen through Holy Week, and Easter itself is a wonderful time to be here. The sea is still cool in April and only comfortable by late May. Beach clubs and sea taxis begin operating through May; the north wind is usually moderate.
June — the best month
Warm sea, long days, every venue open and crowds still manageable. The meltemi begins to establish itself toward the end of the month. If you can choose freely, choose June.
July and August — peak
Everything is at maximum: heat, prices, nightlife, ferry frequency and people. The meltemi blows on most days and hard on some, which cools the air but can close north-coast beaches and disrupt boats. The week around 15 August, the Dormition feast, is the peak of the Greek domestic season; festival weeks such as XLSIOR in late August fill hotels island-wide.
September and early October — the second best
The sea is at its warmest after a summer of heating, the wind eases, and the crowds thin steadily through September. Beach clubs and sea taxis run into early October, then close within a couple of weeks of each other. The first autumn storms can arrive in October.
November to March — the quiet island
A few hotels and restaurants in Chora stay open year-round; almost nothing on the beaches does. Flights and ferries thin out. It is a time for people who want the town to themselves, the winter feasts — Epiphany at the Old Port on 6 January, Clean Monday kites in late winter — and cheap rooms, not for a beach holiday.
About the meltemi
The meltemi is a dry northerly wind produced by pressure systems over the Balkans and Turkey. It blows most reliably from mid-July to late August, often for several days at a stretch. Sailing operators know it well and choose sheltered anchorages; the south coast from Ornos to Elia stays swimmable on all but the worst days.