If it's a "cocktail" like it needs fancy ingredients like bitters or a strainer or something, either the place has the stuff or it doesn't, and you can always tell by looking at the place. And even if you can't, you can always tell if you even want them to make one for you. A shitty bar old fashioned thrown together by some hag isn't going to be worth your money. In those kinds of bars you can still get mixed drinks, like rum and coke or gin and tonic. If you really want a fancy drink, look up "cocktail bar" or "whiskey bar". Restaurant bars will also often have that kind of stuff on hand. It never hurts to ask "can you make me a _?".
My go to used to be a greyhound, which is grapefruit and vodka. Whiskey gingers are good, as are moscow mules, whiskey sours if you're a girl into fruitier stuff, basically anything that looks like one drink mixed in another drink is a safe bet for most bars. My favorite cocktail on the other hand is a sazerac.
