Cocktails to order at bars?

Can't go wrong with an old fashioned - screams I work in finance and don't really know cocktails, but the safe choice and you don't want to stand out as the intern ordering crazy stuff at the bar.  If you want to elevate beyond that, would suggest a Manhattan (Old Fashioned minus the sugar water plus vermouth and bitters) or Negroni (campari + gin + vermouth).  

Paper Plane - this is the way.

Gin and tonic always plays

Macallan 18, neat. That'll show em who's boss

Gin tonic, the only way

- Vodka tonic with lime

- Long Island Iced Tea

- Greyhound

- Jack or Makers and Coke

Definitely recommend Gin & Tonic! A classic cocktail known for its simplicity and refreshing taste. It's like a stylish kimono meets a snazzy bowtie. The gin's botanical blend dances on your taste buds like a graceful dragon, while the tonic bubbles add an effervescent twist. Give it a try!

Blood & Sand if you wanna be cool and mysterious.

Margarita if you wanna be fun.

Corpse Reviver #2 

Negroni is a no-brainer

I agree that Negroni is a no brainer but then it depends if you like this “heavy” cocktail (it is one of my favorite, but still it is gin, Campari and Martini, be sure you can handle that…)

Other cocktails to order/try:

London Mule (Gin and Ginger beer)

Moscow Mule (Vodka and Ginger beer)

Hugo (Spritz with Saint Germain, hibiscus flower liquor)

Boulevardier (a Negroni with whisky instead of gin)

And of course Martini Dry (Martini and White Vermouth)

Good suggestions all around, hard to go wrong with gin/vodka tonics/sodas.  I mostly prefer whiskey drinks but a few of my favorites not mentioned here are:

Sazerac - Similar to an old fashion but with rye whiskey instead of bourbon.  Typically if I order either I opt for Bulleit Rye if they have it.

Gold Rush - Names may vary, but this is a variation on the Bees Knees (Gin, lemon, honey), but with whiskey instead.  

Penicillin - Scotch-based with lemon and honey syrup.  Crisp and refreshing.

Tom Collins - A great summer cocktail and good alternative to a GnT.  Gin, lemon juice, simple syrup and club soda.

Vodka soda or tequila soda because they're less caloric and more macro-friendly than most drinks. 

dirty martini

old fashioned or Manhattan.  

Back in the day I'd see financiers ordering drinks shaken, not stirred, among other top shelf single malts. 

Now I see financiers ordering White Claw and Smirnoff Ice.  And it's sad.

Negroni or an old fashioned

whiskey sour, old fashioned or sazerac, a highball of some kind in the summer or a mojito in a rocks glass, maybe a mint julep

Usually a negroni or boulevardier depending on whether I’m feeling gin or whiskey. You can also never go wrong with an old fashioned. Have also been enjoying a bensonhurst in recent weeks.

If you're a hard-core alcoholic, it's either an Olde English 800 or Mad Dog 20/20.

Can’t go wrong with either tequila or whiskey on the rocks. 

Dirty vodka martini with blue cheese stuffed olives or GTFO

Whatever you end up ordering, be a little careful, especially if you are at a "fancy" cocktail place. Which I imagine is the only sort of place an MF would hold happy hours at.

Cocktails can have a lot of flavours masking the alcohol, making it easier to have bit more than you thought you did. And fancy cocktail places tend to put more flavours in, plus are more likely to give you drinks that actually have a decent amount of booze in them as opposed to dives that stingily measure out each ounce of liquor.

Moscow Mules are good if you like vodka

Negronis are safe

Not a fan of whiskey so those are usually the two I default to.

If you’re a high test male, straight whiskey

otherwise feel free to dilute with cheap soda/coke

I'm sorry, but where do you work?  You said "interning at a MF" but I'm trying to figure out what MF stands for.  Given that your bio says AM, I'd almost want to say Mutual Fund, but nobody says they work at a mutual fund.  It's common to say that you work at a fund shop, or you work for an Asset Manager, but I've never heard the phrasing that you used.

Anyway, to answer your question:  Anything that would be reasonable at a country club would be reasonable at a happy hour in AM.  A G&T or Aperol Spritz are easy lighter summer options.  In colder months or after a bad day an old fashioned or Manhattan are also pretty good, but be careful, those are all booze and can knock you on your ass pretty quickly.  You're also fine with a martini (Gin or vodka, none of these sweet things) and a cosmo is borderline OK, but under no circumstances order a long island iced tea. or a rum/jack and coke.

Also, an even bigger rule: if it's a happy hour don't get drunk.  There are certain circumstances like the Christmas party where it's OK, but a happy hour isn't one of them. (and oh man, do I have Xmas party stories--There was this one year when one wholesaler socked another in the face--They both agreed that nothing had happened the following morning, because neither wanted to lose their jobs, but the guy had a shiner for almost two weeks)

Vesper Martini

Seriously, that’s what you get.

Vodka water without ice. This is what the upper class really drinks.

Gin Rickey if you want a break from a G&T. Super refreshing and bright with citrus and mint being the primary flavors

Gin and Tonic or Vodka with Sprite Zero.
Simply and to the point.

Scotch on the rocks.

It's the historical standard for bankers, what everyone ordered when out, and what many offered in the office. For generations. Simple and classic.

I agree with the comment above about sticking 2 ingredients. Personally, if you want to give a strong first impression, I would go with vodka water.