Just about everyone agrees that the best time to visit is Greece is spring and early summer (mid-Apr to mid-June) or autumn (Sept to mid-Oct). This way, you'll avoid the summer high season, with its inflated prices, hordes of tourists and high temperatures (heat waves of 100°F/+40C are routine). In fact, unless you really like scorching heat, crowds, and overbooked planes, ferry boats, and hotels, August is to be avoided. These drawbacks loom especially large if you plan on visiting some of the more popular islands, Mykonos and Santorini first and foremost among them. In the spring, you’ll see more wildflowers than you could have imagined—and swim in a sea that’s a bit cool but more pleasant than you had hoped for! In the autumn, you will enjoy golden days with still-warm waters. One drawback: Off-season there are fewer boats and flights to the islands, where some shops, hotels, and restaurants do not open until June and then close in October. During the off-season life comes to a standstill on many islands, or at least turns its back to tourism, and wintertime rains can dampen any romantic notions of lonely wandering in empty landscapes.
Something to consider if you are coming to Greece in the spring: During Easter week, nearly every hotel room outside of Athens is booked well in advance by city Greeks who head to the country to celebrate Greece's most important holiday. Many sites and museums are closed Good Friday, Easter Saturday, and Easter Sunday, while many shops close on Good Friday and Easter Saturday. And when St. George's Day (usually celebrated Apr 23) falls in Lent, it is celebrated on the Monday after Easter Sunday, which just prolongs the Easter break. Also, the Feast of the Virgin on August 15 is an enormous holiday, especially on Tinos and Paros, but also on virtually every other Greek island and across the mainland.
I've been visiting Greece since the 1960s, and for the last decade I've spent about half the year in Greece, most of it in a village in the Peloponnese, south of Athens. The one thing I can swear to is that the weather is getting less predictable every year! When Greeks talk about the weather, increasingly they say that everything is ano kato (upside down). Some things everyone agrees on: the winters can be chilly (sometimes with unusual bursts of warm weather, then again, sometimes it even snows in Athens). It can go from warm to downright numbing. Many buildings are not insulated, and the centrally controlled heating is often intermittent, making the cold season seem very long indeed. Summers are just plain hot (and usually dry), sometimes reaching 110°F (43°C). As the saying goes, only mad dogs and Englishmen would venture out in the midday sun, hence the siesta between 3 and 6pm. The seasonal north (Etesian) winds blow mid-July to mid-August, but it can get very windy anytime, stopping ferry transport.
Every day in Greece is sacred to one or more saints. That means that every day, at least one saint (and everyone named for that saint) is being celebrated. Tiny chapels that are used only once a year are opened for a church service followed by all-day wining and dining. If you're lucky, you'll stumble on one of these celebrations.