Depends on what point in the past you're talking about - and what social class, of course. The Greek writer Herodotus in the 5th century BCE is generally considered the "father of history", for weaving together disparate stories into an overarching factual progression narrative.

The Romans themselves were quite fond of history, maybe even moreso than the Greeks they learned the discipline from - a lot of Roman writing is not only about history, but specifically about skepticism of their own history. They discuss sources and myths and alternate interpretations - it's very good. Of course, those writing books would be upper-middle class at the lowest, and most would have been genuine aristocrats whose net worth would make any of us plebs blush.

Sparta, in fact, was revived specifically to cater to wealthier Romans seeking to 'see the history', a kind of prototourism. That it was probably inaccurate was of less concern. Julius Caesar himself broke down in tears, because he realized that at Alexander the Great's age, he had already conquered an empire, while Caesar was still a nobody at the time - which clearly demonstrates an understanding and appreciation (or envy) of the history of the Macedonian Empire at the least.

Graffiti in Pompeii suggests that ordinary urban citizens in the Empire, or at least in Italy and its colonies, possessed a reasonable standard of education.