Adana is in the hottest part of mainland Turkey and the summer heat is no joke — July and August routinely top 35 °C and the air is heavy with humidity from the irrigated cotton plain. Unless you have a specific reason, avoid those two months.
April to early June is ideal: warm but not punishing, with citrus blossom in the parks and roads clear of the school-holiday traffic. October and November are similarly comfortable, with a refreshing breeze from the Mediterranean. Winters are mild (10–15 °C in the day), occasionally wet, and a fine time to visit if you also want to combine the city with a beach run down to Mersin or the ruins at Hierapolis-Cilicia.
If you can, time a visit to coincide with the Altın Koza film festival in late September or the orange-blossom Portakal Çiçeği Karnavalı — Citrus Blossom Festival — in early April, when the city fills with parades, jazz concerts and a serious appetite for bici bici.