One source would be folklore of course, people were telling tales in all ages and countries.

Another source would be elder generations telling their experiences, it is practically certain that Vorenus' father or grandfathers would have served in an army fighting with or against Sulla, and of course his mother and grandmothers would have their own experiences about Sulla's dictatorship, Marius' short but bloody seventh consulship, the occupation of Rome before.

Written sources that the average Roman Joe would be certain to be familiar with are inscriptions. Romans always tried to come up with witty and easy to remember inscriptions for their grave, much like people in our age. Lists of magistrates would be also inscribed to stone, and were on public display. Laws were also written down.

I am unsure if inscribing laws into stone and putting them on public display was a practice for every single law, or just some, but it is certain that written records were made from all of them. Polybius writes that he got to see a bronze table in the aedils' treasury on the Capitolium which contained the alliance betwen Rome and Carthage dated for the first year of the Republic. So there were extensive historical records available. Vorenus is reasonably wealthy, I think IRL he would come from a knightly family, or at least from the first class, so he could probably ask and read at least some of these records.

Further inscriptions would be on every single temple and public building. Roman aristocrats wanted to get the value for their money, so if one built a temple on their own expense, they were sure to put their name on that temple.

Prominent Romans could have statues, and of course they also had descendants. These descendants and relatives were sure to boast of the deeds of their forefathers. They would try to put their faces on money, and it was customary to have busts of their ancestors at home. So a proper Roman aristocrat would know the names and faces of their ancestors, and the older generations would teach them their legacy of course. Vorenus has probably served with many aristocrats. IRL Lucius Vorenus is mentioned as a high ranking centurion of the IXth legion under Quintus Cicero, so I imagine he would have been on first name basis with Cicero at least. (He was the brother of Marcus Tullius Cicero by the way.)

And of course there were other public records that the magistrates would make, and there were historians who have written books leaning on these sources. The first of them was Fabius Pictor in the III. century, so by the time Vorenus was born there was quite a selection of historians to choose from.

But I have to add that Lucius Vorenus is not your average Roman Joe. As said, IRL he was a high-ranking centurion who probably came from a knightly family or became a knight during his life at some point (since the census was based on wealth and Caesar's campaigns in Gaul were lucrative), and in the series he is also shown as a reasonably wealthy and influential member of Roman society.

For reading more I would recommend Tim Cornell's The Beginnings of Rome, he has a whole chapter dedicated to the sources that might have been available for the historians that we use as sources.

Edit: Of course the Cicerones were homini noves in the sense that Marcus was the first of the family to become a senator. But they were still very much part of the Roman/Italian aristocracy, a wealthy family of high status. They just did not live in the capital, but in the smaller Arpinum. Much like Maecenas could be said a "new man", while he was the descendant of an ancient Etruscan dynasty that has ruled his hometown for centuries.