April to early June and September to October are the best windows almost everywhere in Turkey. Temperatures are comfortable, the coast is warm enough to swim by late May and still warm in October, and the archaeological sites are bearable in a way they simply are not in August.
July and August on the Mediterranean coast regularly pass 35°C, and inland Anatolian sites such as Ephesus and Hierapolis offer very little shade.
Istanbul works year-round because it is a city break rather than a beach trip. Spring and autumn are mildest; winter is grey, wet and occasionally snowy, but the city is at its emptiest and hotel rates fall sharply.
Bear in mind the daylight. In late December sunrise is after 08:20 and sunset before 16:40, so a winter day gives you barely eight hours of light for sightseeing.
Cappadocia sits at around 1,000 metres and gets properly cold, with snow from December to February — which is genuinely beautiful over the fairy chimneys, if you can stand the temperature. Balloon flights are weather-dependent and are cancelled more often in winter.
Eastern Anatolia, including Van and the Kaçkar mountains, is effectively a summer destination. Mountain passes there can close in winter.
November to March, outside the New Year period. Coastal resorts largely shut down, but Istanbul and Cappadocia stay open and are much cheaper.
Roughly late May to late October on the Mediterranean coast, and a slightly shorter window on the Aegean.
Year-round at sunrise, weather permitting. Cancellations are more frequent in winter and in high wind.