At a barber.
A tonsor would have been identified in a commercial area by a display of bandages, which over the centuries evolved into the modern barber pole. Tonsores also shaved. (In the Empire, when beards were fashionable again, they groomed beards.). Barbers also supplied men with hair product, always their own blend (usually olive oil-based pomades, scented).
Varro provides the oldest citation of barbers in Rome, and they were a common enough feature in comedy.
The shears they used were connected, like one flat piece of metal bent to form a spring-like shear, with the ends turned and sharpened. Plenty of examples found in archaeological sites.
A quick search turned up an old article from 1891 by F. Nicolson. This is exactly the kind of question scholarship from that era sought to answer and is still valuable for.
Hygiene and grooming was very important to Romans, especially those living in towns and cities. The more money one had, the more one spent on it.
Great question! Too often scholars focus on literary criticism or monumental archaeology. But these kinds of things offer an insight into daily life.
Rome would feel more modern to present-day people than the Middle Ages. Their fashion sense would be surprisingly accessible and understandable to us.