Turks generally prefer to drink Turkish coffee after the meals, mostly after dinner.
Firstly you should have enough time for Turkish Coffee. It's a not a coffee to drink on the go. Mostly suitable time and surrounding would be as such, sitting comfortably with your friends or family and don't have any time constraint.
Presentation is one of the most important part of the ceremony. You never drink Turkish coffee from a cartoon or a glass but you require special small coffee cups for that. Options to serve Turkish Coffee with would be; chocolate, Turkish delight or liquer.
Turkish coffee generally comes with a glass of water.Before having a sip from your coffee you should drink the water, not all as much as you want. The purpose of water is to clear the aroma in your mouth so you can enjoy the taste of coffee more. It's already a small cup so sip slowly. My husband treats it like it's any other coffee and generally orders the second one within 10 minutes, that's wrong:) it's meant to be drunken small amount like espresso.
Last but the least, actually more important aspect of drinking Turkish coffe, never try to finish it all! Don't mix it with espresso. At the bottom of the coffee cup there are coffee grounds, eating them would take all the joy and taste of drinking Turkish Coffee.