Did you just go to a barber and give instructions more or less the same as today? Based on statues and drawings, people clearly got haircuts, so how did it work? Were there North Korea style preset cuts?
Depends on who you were, really! Let's start at the top and work our way down.
The top, in the Roman Empire, was the Emperor. There are a few interesting hair-related anecdotes concerning emperors, usually by Suetonius who gives some great physical descriptions. A couple, though, discuss haircuts. Here's one about Augustus:
He was unusually handsome and exceedingly graceful at all periods of his life, though he cared nothing for personal adornment. He was so far from being particular about the dressing of his hair, that he would have several barbers working in a hurry at the same time, and as for his beard he now had it clipped and now shaved, while at the very same time he would either be reading or writing something.
Obvious flattery aside, it gives a small idea of how the imperial household handled the appearance of the Emperor - with extreme care and precision, as well as speed. This, apparently, was Augustus' manner on all but the most disastrous of occasions. After Teutoberg, for example..
they say that he was so greatly affected that for several months in succession he cut neither his beard nor his hair, and sometimes he would dash his head against a door, crying: "Quintilius Varus, give me back my legions!"
Considering this was near the end of his life, you can only imagine a wild-haired old man, furiously spurning his servants, etc etc. But other emperors also show specific quirks, such as Nero apparently putting his first beard hairs into a small golden box and offering them to Jupiter.
The wealthy, too, would simply buy a slave barber, who would ensure that they always looked their best. These barbers would take care of ensuring that their hair was properly styled according to the fashion of the day, ensure that they were clean shaven (or not as per, again, the fashion). If having a smooth face was in, then the barber would not only shave the person with a razor, but would pluck stray hairs with tweezers, followed by rubbing the face with pumice. Still works rather well today if you want to use it, and it's apparently quite good for both depilation and exfoliation. Other methods were available, such as "waxing" with pitch and singeing off hairs (think laser removal except far less precise), and would be readily available, especially to those with funds.
"But Celebreth!" you might say, "What about the people who weren't part of the .001% and couldn't afford slaves in a mansion?" Well, dear reader, that's where the fun part starts. Let's talk about barbershops! So barbers were brought into the Roman world (apparently) in ca. 300 BCE, when a Roman (Publius Ticinius Menas) brought one in from Sicily. Varro makes the spurious claim that there were no barbers before that, "as old statues show, since they have long hair and long beards." Take that as you will. It appears, though, that the fashion of barbers took off as the city grew in wealth and prestige. Pliny claims that Scipio the Younger was the first Roman to shave daily, which is a definite start to ~200+ years of clean-shaven Romans. At least the wealthy. The lower class couldn't necessarily go with this fashion statement - Martial has a great poem that makes fun of a guy with a huge beard. I'll quote the whole thing 'cause I love Martial (adapted from the Loeb translation, cause I like most of it, but not all of it):
It's winter and harsh December stiffens, yet you, Linus, dare to hold up everyone you meet in Rome, whether they come from here or there, with your snowy kiss. What are you able to do that would be more hurtful and cruel if you had been stabbed and flogged? In this cold I would not have my wife kiss me or my innocent daughter with her coarse lips. But you are sweeter and more elegant, with a discoloured icicle hanging from your canine nostrils and a beard as stiff as a Cilician barber with upturned shears cuts from a Cinyphian husband! (goat joke) I fear less to fall in with a hundred cunt-lickers ("cunnilingis") fresh from action. So if you have any sense and shame, please hold off your winter kissing, Linus, until April.
This fashion was taken pretty seriously among the upper class in general - Livy notes that a man who was previously exiled was forced to shave his exile-beard clean before he was allowed to do anything administrative, even attending the Senate. Having a beard and wild, untrimmed hair was either a mark of poverty or a mark of grief. But hey, not everyone in Rome was plum broke - barbershops certainly existed. So what would you do if you weren't as broke as poor Linus up there, but also not wealthy enough to afford a house and staff it with slaves? Well, you'd go to the barbershop. A barber would have a certain number of basic tools, many of which people today would not be unfamiliar with: a comb, mirror, shears, curling iron, tweezers, razor, and knives for nail-cutting. Some barbers were travelers, but the more successful ones had shops - which, as you might imagine in a city plagued with traffic problems, became rather crowded sometimes.