Like someone earlier said, if I'm at a nicer bar/restaurant, I'll usually order something off the cocktail menu. If it makes the menu, then it's something they do well and it's going to be consistent from bartender to bartender. Here at least, drinks can be very specialized. I love lemon drop martinis, but only from a few bartenders--most of them are too sweet. If I'm ordering a LEMON drop, then I want that tartness. I should probably be ashamed that I know individual bartenders
From my favorite nicer bars in the city:
French 75:
Gin, lemon, simple syrup, and a splash of champagne. The classic French 75 actually uses cognac, and there are a few bars here that give you the option, but I actually think the gin is tastier, but only in this drink. I normally don't care for gin.
Kir Royale:
crème de cassis, champagne, orange peel
Sometimes if you order a Kir Royale, they'll use Chambord instead of cassis. This is technically a Hooker's Cocktail (at least in NOLA). It's a really good drink too, but if you specifically want the blackberry flavor instead of raspberry, you need to specify that you want crème de cassis.
I also like a French martini, with a splash of champagne. I like just about anything with a splash of champagne.
If you're ever in New Orleans during Christmas, go to the Sazerac in the Roosevelt. They have a Candy Cane martini that will knock your socks off. The ingredients sound like it should be a total sugarfest, but it's not.
If I'm out and about in a new bar/city OR the bar is packed, I order a vodka tonic. Hard to screw up a vodka tonic. Unless they put soda instead, but I like that too, cause you know, vodka.