As many have already said, only the privileged classes would go out of their way to learn historic things (perhaps monks and nuns might have access to written histories, or someone wealthy enough to be literate might read something like Herodotus). But most people told histories through myths, legends, rhymes, and ghost stories, often for entertainment.

Aboriginal people in Australia pass(ed) history down as oral tradition, and my shallow understanding is that their histories are incredibly detailed and lengthy, and the teaching is extremely rigorous and rigid, so they are passed on exactly as before; they apparently map well to natural history going back an astonishing amount of time.

A farmer in what is now rural England may perhaps have passed on legends about local history, or may have learned about it in song. Medieveal Europeans often memorialized notable local royalty through song and legend, such as "Good King Wenceslas" or the story of Lady Godiva. Traveling entertainers or troupes might tell fun stories, alongside actual historic recountings, in verse. That's partly why so many poets in centuries past wrote their epic stories in rhyme: there was a tradition of people, likely illiterate, memorizing and reciting the entire thing for audiences. The practice of history through song was something people took up again during the labor movements of the early 20th Century in the US; as the stories and histories were suppressed officially, people wrote folk songs about coal town battles, union leaders, awful bosses, and what life was like for workers. And, of course, a lot of history that people learned was religious.

In Tibet people have been passing histories of monks and monasteries down for a long time but I'm not sure how far it goes back. Because of the tradition of locating the reincarnation of a specific monk or teacher in every generation, naturally the history of that person through the centuries would be relevant. So, the stories of monks and monasteries was one possible subject of history.

A look through Shakespeare's oeuvre shows that in the late 1500s people in London were watching plays about famous ancient Romans and famous English Kings. In the late 1300s Chaucer was also writing Classical histories (mixed with mythology) and histories of notable nobles and church officials (not always complimentary) in verse. His audience would have been noble men and women, as well as the rising upper middle class people and merchants.