There was a very official event for a Roman boy's coming-of-age. We don't know everything about it, and no one source fully describes the proceedings, but we can piece it together from scattered mentions. The composite we get is therefore not representative of one particular time or place in "Rome," but we do get a general idea. And, of course, this pertains only to free-born Roman males (sorry, ladies), and typically upper-class.

The ceremony had no set age, but typically occurred when a boy was 15 or 16. Cicero was 16 when he "came of age," as was his son and nephew. The future emperor Augustus was 15, as were his two grandsons. The second emperor Tiberius was 14, and Caligula was 18. It is never specified why or when exactly the event was triggered. It might have been a matter of planning or convenience, or timed to coincide with some other festivity. Some hypothesize that it was tied to the kid's first signs of puberty (whatever that might have been), or first sexual exploration (again, whatever that might mean). No one knows for sure, but there are many theories. Personally, I suspect there was a religio-superstitious aspect, because superstition ruled every aspect of Roman life. All our evidence points to the father as the one deciding the time, and so perhaps it was just whenever the father thought the kid was "ready."

On the chosen day, the boy would come before the Lares (household spirits) in their little shrine in the family home. There he would ceremoniously remove and offer his bulla, the little pendant medallion worn by every free-born boy to ward off ill fortune (child mortality rates in the ancient world were no joke). He would then lay aside the garment which was the uniform of an adolescent: the toga praetexta, a white toga with a red stripe around the bottom edge. The bulla and toga praetexta together were the insignia puerorum ingenuorum, the visible markers of youth. Their removal was a symbolic casting-off of that phase of life. The bulla certainly, and possibly also the folded toga, would remain at the shrine of the Lares, perhaps to receive further propitiatory offerings by the young man in the coming years.

Next, the boy put on a special shirt, the tunica recta. This was a "good-luck shirt" woven on a special loom (yep). Brides also wore the same tunica as a nightie on the evening before their wedding day. There is some sort of sexual fertility aspect going on here, but the specifics escape us. After the good-luck shirt, we move on to the heart of the ceremony: the boy receives, from his father's hands, his new garment, the toga virilis ("manly toga"), a pure white toga with no red banding which was the marker of a free-born adult male citizen.

This was a big deal, and the atrium of the house would have been packed with family. Numerous inscriptions record commemoration of the event ob honorem togae virilis, even by the young man's slaves. As he became a man, they too got a sort of promotion into a new phase of life, and could expect their duties (and horizons) to change significantly. The ceremony itself is not difficult to understand, anthropologically speaking: much like a reptile, the young man sheds his old "skin" and puts on a new one. It is the typical ancient symbolism of passage across a liminal space, from one domain to another. Certainly the Lares were "observing," but also numerous other gods like the Cardines, spirits of the threshhold, Iuventas, goddess of youthful exuberance, and, in some capacity, Mars, into whose arms the boy would soon be walking.

The ceremony continued with the transition of the boy to man, from inside the house to outside. The little party processed from the atrium into the street and on to the public spaces of the town: the forum, and the temples of Rome's chief civic gods (Jupiter, Minerva, Juno). In Rome, it was the ad Capitolium ire "the going to the Capitol." Later there is some evidence that it became fashionable to go to the temple of Mars Ultor instead (built by Augustus). At the temple, the boy-turned-man made sacrifices (probably) to Jupiter, as well as other various rituals. It was probably a common sight day to day. One probably notices a good deal of similarity with the Quinceañera.

All that so far, and no mention of the beard. This is because there was another, separate ceremony for shaving, also a coming-of-age, but with some distinct differences. First, the assuming of the toga virilis was not tied to a kid's physical state (at least not entirely), but rather his legal and societal state. The changing of togas signaled his entry as an adult into the adult Roman world of the forum, courts, altars, battlefield. We only know of the beard ceremony from scattered sources, and there not very many of them. The most notable are Petronius Satyricon 73.6, where we learn the ritual is called the barbatoria. We also get a description of it in Suetonius's biography Nero (12.4), where we hear that the ceremony involved (of course) a sacrifice, the ritual placing of the shaved beard hairs into a little box, and their deposition as a votive object on the Capitol. That was for the fancy future emperor: most people probably just dedicated them to the Lares in their own house. We get the impression that this ceremony happened much later than the toga ceremony: Augustus put on the toga virilis when he was 14, but did not shave his first beard until he was 23 (Dio 48.34). Other authors also make it pretty clear that the barbatoria was separate from the toga ceremony, and probably later (Juvenal Sat 8.166, e.g.).

Therefore: yes, we have to assume that boys would not shave until in their late teens or early 20s. Their downy growth perhaps marked them out as a subcategory of manhood: new to the toga virilis and not yet fully grown, but in an in-between phase. It is also possible that the beard-shaving ceremony was tied to some other, unknown milestone. Interestingly, the barbatoria did not seem to be limited to free-born citizens. It was "available" to all males. One thing to note: we must keep in mind that ancient Romans as a group were a mix of mostly Italic, some Celtic, some Greek, Iberian, and other Mediterranean peoples, and as a general rule were not quite the hairy Germanic-descended barbarians that now occupy the majority of Europe and the United States (they were also half a foot shorter, on average, than modern Western males). Though the Antonines sported some pretty impressive beards, at least in sculpture busts, the typical Roman male would not have been able to muster a big bushy Viking beard at age 16.

For a comprehensive overview of the toga virilis ceremony, with numerous citations, see Roman Dress and the Fabrics of Roman Culture ed. Edmondson and Keith, 2008, especially Ch 2 by Dolansky. See also Wiedemann, Adults and Children in the Roman Empire (Yale 1989); Eyben, Restless Youth in Ancient Rome (Routledge 1993); and the body of work by Marc Kleijwegt.