June and September are the best months for most visitors: the Adriatic is warm enough to swim, the roads and beaches are not yet (or no longer) at peak capacity, and accommodation prices are significantly lower than in July and August.
July and August are when the island is at its busiest and most expensive. The main beaches around Novalja and on the eastern coast are packed, and the Zrće beach clubs are operating at full intensity. If that is what you are coming for, this is the right time; if not, avoid the peak.
Outside summer (October to May) the island is quiet, some businesses close, and the landscape takes on the stark, windswept quality that defines Pag in its non-tourist face. The bora wind is at its strongest in winter and can be genuinely severe. The cheese and olive oil are produced year-round and the island is worth visiting for those who enjoy off-season quiet, but most organised tourism is suspended.
The Pag Carnival in February is one of the most traditional in Croatia, with distinctive masks and costumes; a good reason to visit in the dead of winter if you plan carefully.