Diyarbakır is a socially conservative city and alcohol is less visible than in Istanbul or Ankara — you won't find a bar on every corner in Sur. That said, tea houses (çay ocağı) are everywhere and are the social centre of the old city. Locals will nurse a glass of strong black tea for an hour over a backgammon board; joining in is perfectly acceptable and often warmly received.
The area around the Hasan Paşa Hanı has a cluster of cafes in the old caravanserai courtyard that are pleasant in the evening. Coffee is increasingly available alongside the traditional tea; the newer cafes in Yenişehir (the modern part of town) follow mainstream Turkish cafe culture with good espresso drinks.
For a cold drink on a hot afternoon, şalgam (fermented turnip juice, sharp and purple) and ayran (yoghurt drink) are both local staples and better than anything you'll get from a vending machine.