Greece is a hot place in the summer, so light clothing is in order. The temperature sometimes reaches 45° C for stretches of four to seven days. This occurs usually mid-July until mid-August. For most of the time during this period though, the thermometer hovers between 34° and 40° Celsius. Shorts, T-shirts, and bathing suits are a must, as is a variety of head gear and sunglasses. It would be almost impossible to stay outdoors for any length of time without any hat, and it might be hard to negotiate harsh terrain in archaeological sites without athletic shoes.
You might also want to pack a sweatshirt or a long sleeve shirt. It comes in handy in an air conditioned environment (like in ferry boats or airplanes), and in the evenings on the Greek islands where the temperature dips a bit bellow comfort levels when the "meltemia" winds blow over the Aegean sea.
Also keep in mind that although it rarely rains in Greece in the summer, you should be mildly prepared for the odd major rain storm that might strike at any time. Usually such rain storms last one or two days, but on rare occasions can last longer.
Many religious centers and monasteries do not allow men with shorts and women with trousers, mini skirts, or shorts, (or anything that reveals too much skin) to enter. A shawl can be helpful to cary around since it can be deployed to cover the shoulders or to be tied around the waist over shorts.
You can expect cool weather, heavy winds, and frequent rain after mid-September.
Greek weather is mildly cool and a sweater is a necessity, especially at night. It is possible to swim until the first half of October in most places in Greece. The sea is warmer further south towards Crete and comfortable to swim even in October. During the Fall it rains more often, though not every day, and most days might be still warm. Pack long trousers, a couple of sweaters or sweatshirts, an umbrella and a raincoat, but you might also bring along a bathing suit just in case.
It does get very cold in the winter which lasts from late November until early February. During this period in rains often and the wind is bone-chilling. Further north, and especially in the mountainous regions of Epirus and Macedonia snow is a usual occurrence. Heavy clothing is advised during this period.
Pack a heavy coat, long trousers, heavy socks, boots, an umbrella, and you will not need a bathing suit.
Since the Greek winter is considered "short" and "mild", most people fail to realize that Greek dwelings (hotels, rooms, homes, etc.) are not prepared for heavy winter cold. The heat and insulation is usually inadequate for the frequent dip in temperature - especially during the night hours. You don't have to pack blankets, but having warm sleepwear is advised.
The period between February and May is unpredictable, with most days being very warm. Rainstorms are not unusual, but most days are dry and very pleasant. Light clothing is in advise, with a raincoat thrown in just in case. The sea is a bit too chilly for swimming during these months.