No love for the “Adios Motherfucker?”
Kidding, that is for the hardcore (me). You seem for the most part to enjoy vodka/sour drinks so I might offer up a Kamakazi - vodka, lime juice and depending on the place either Triple Sec or Cointreau. (Depending on where you are this will be served either as an “up” (martini glass) or a shooter. BTW, and I might be alone on this - there is no shame in sipping a shooter.)
Some establishments are known for beefy Brunch Bloody Mary’s or Clamdigger’s, try to sample them if they make a particularly good oone. An old fashioned oldfashioned with muddled maraschino and orange and some bitters with a rye whisky is nice to try.
I’ll recommend the sidecar as well - sort of the great-grandpa of the margarita, with brandy, grand marnier, and lemon juice.
The Singapore sling is also a good one, but it might be hard to find a bar that has all the right ingredients (Heering and Benedictine are both uncommon and pretty expensive).
If you’re in the mood for a hardcore drink, go with a zombie - four or five different kinds of rum, and possibly some fruit brandy mixed in. They call it that because it turns you into one. 
If you like your drinks dry, make sure you get your Old Fashioned made the RIGHT way - rye, sugar, water, and bitters. Muddled fruits are for lady drinks.
Sorry, but by law, you can’t recomend a Singapore Sling and still get to call an Old Fashioned with fruit a lady drink
Yeah–if the bartender is relaxed (and especially if I know them), I’ll say something like, “I don’t know much about cocktails, and I like trying new ones. What’s something you’re really good at?” I might add details if I’m in a certain mood (“Do you do any hot drinks? Do y’all make your own sour mix, and if so, could you do me something good and sour? What’s something tasty you can do with some bitters in it?”) but then usually I’ll go along with whatever they want to make. In my limited experience, most bartenders are delighted to be given free reign like this.
Christopher Lee offers some suggestions.
I should probably be running, right?
I just made a bourbon sidecar last night when I got home from work. It was very tasty, but nothing beats my old fashioned.
What kind of martini did you have? If you had it with vodka, you didn’t have a martini. (Yes, I’m a gin girl. Love gin. Use Hendrick’s, you’ll thank me. And do have them add at least some vermouth - it’s not properly a martini without it.)
Old Fashioneds are currently my go to. We started the summer of the Manhattan this summer, but it didn’t really take off. Not sure why.
When I ordered the martini, I told the bartender I never had one before and wanted just a regular plain old martini. I wanted to make sure I wasn’t getting some alternative martini like appletini, etc. It tasted like gasoline and I don’t like olives. Maybe I’ll give it another try some day.
I came in here to post this. I had never heard of it until today and it sounds good.
Personally I like a good dry Gibson, but seeing as you didn’t like a martini I doubt a change in garnish will make much of a difference.
Not so much a “cocktail” as it is a shooter but a round of Irish Car Bombs is always great fun.
Is ginger beer commonly stocked at bars?
Obligatory Futurama quote
Bender: “Hey bar keep, I’ll have a fuzzy navel and she’ll have the girliest drink in the house.”
Bar Keeper: “Two fuzzy navels coming up!”
I disagree.
No cocktail whose recipe starts with “8 parts gin” can reasonably be called a lady drink.
Whatever you order, avoid “rack” versions. The quality of the booze is usually one step above paint thinner. Always ask for a “mid-shelf” quality, at minimum. For example, don’t ask for a vodka tonic, ask for a Stoli and tonic or Ketel One and tonic.
And they’ll reach back to the back of the shelf to play with things they don’t often get to play with. We discovered Canton ginger liqueur doing this - and its now a staple around here - in apple cider warmed up, with bourbon, in pear martinis, in lemonade.
And a plain ol’ martini tastes like gasoline with an olive. Its gin - just really cold gin. In the early ages they used to put enough vermouth in it for it to change the taste of the gin, but nowadays, the vermouth is there only to turn it into a martini. And unless you order your martini dirty, the olive should be subtle. So there isn’t much difference between a martini and doing ice cold shots of gin.
I’m a fan of the girly cosmopolitan - which is a vodka martini with orange liqueur (makes it sweeter) and only enough cranberry juice to make it pink. If you taste CRANBERRY, you are doing it wrong. That’s the take for most people with martinis, if you can taste vermouth, your doing it wrong (and vermouth itself sort of adds a lighter fluid taste to the gin - I’m not a vermouth fan).
I like my old fashioned with bourbon. I use one peel of orange and one peel of lemon. Squeeze the peels over the glass so that the oils hit the inside. You can rub the depleted peels on the glass rim for a little extra but I don’t usually find it necessary. The bourbon, ice, a touch of simple syrup and a drop or two of bitters.
Hmm… It’s 5 o’clock somewhere, right?
Just had 2 of these Friday night with Maker’s Mark
Yum.
Tequila Sunrise and Brass Monkey are my two favorite cocktails.
Brass Monkey - I like the version with rum, vodka and orange juice.
I like my drinks to taste like drinks, so I like Martinis, Mojitos, Caipirinhas, Manhattans and old fashioned Daiquiris.
I’m trying hard to find a unifying theme among your favourites, but you have 3 very different types of drinks.
Rusty Nail- Scotch with honey liqueur
Margarita- Tequila and citrus
White Russian- Vodka, coffee and cream.
I’m thinking of things from the same families which might appeal.
White Russian- a-likes:
Brandy Alexander
Mudslide
Baby Guinness (Kahlua with a Bailey’s layer floated on top)
Margarita- a-likes:
Old Lay- A Cadillac Margarita with Grenadine
Tequila Collins/ Juan Collins- a Tom Collins but made with Tequila.
I think the Daiquiris and Caipirinhas would fit in here.
You are on your own with Rusty Nail-a-likes- I can’t stomach Drambuie.
Żubrówka (Bison grass vodka) with apple juice and alcoholic ginger beer is a nice twist on the classic Tatanka, if you can get it.