This is a good question that a lot of people wonder about. Ordering whatever you like, as many of these answers say, is totally right, though I don't think that's the full story. The way that you order can help you build a relationship with your bartender (and other patrons), and that is always a good thing. An example of this is a shot of a liqueur called Fernet Branca, a shot of Fernet for a while now has been known as the bartenders handshake (don't call it that when ordering, just ask for a shot of Fernet), it's a way to tell a bartender that you respect their craft. You can get any ol' beer to wash it down. Keep in mind though that Fernet tastes pretty terrible to most people the first time they try it.
The biggest thing I would say is to try new things, get to know what's out there. This will help you learn what you like and let's bartenders really flex their skill, and they love it... at cocktail bars, not so much at dive bars, but it sounds like you have a nice spot there. If you like Gin, google some classic gin drinks and try ordering them. Order an Army Navy, a Corpse Reviver #2, or a Hemingway Daiquiri for example. Keep in mind that you won't be know the difference between the drinks that your bartenders know and the ones that are obscure, until you get to know the lay of the land so just stay humble and keep to the really familiar drinks when the bartender is crazy busy. Also, feel free when it's not busy, to use adjectives to order... "I want something sweet with whisky" or "something bitter and herbal". Keep the communication open, they know what they're doing.
Just to get the ball rolling for you, here are a few to start with....
Brandy Alexander is sweet and the perfect nightcap. An Amaretto Sour with a splash of bourbon, another easy going cocktail, it's a twist on a classic and created by one of the biggest names in the industry. A Negroni, it became super popular not long ago and has been a little bit over done since then but, you should order one and know what everyone went crazy over. Be ready, it'll be bitter.
As a last note, whoever said that you should avoid girly drinks as a man is wrong, there's no such thing. I've proudly ordered a Cosmopolitan in a bar and it was delicious. Sit upright and hold your head high. If you're looking at me I'll give you something to see.