This is called the trivium, the lower three of the seven classical liberal arts (paired with the quadrivium of music, arithmetic, geometry, and astronomy). It is where we get the words "trivia" and "trivial" from, because they were presented first to students, were overall easier, and involved a lot of memorization of things that experts had written.

The answer to your question is genuinely not that complicated: it's the same stuff you'd find in a modern grammar book for Latin or Greek, plus a fitting vocabulary and probably also literacy in most cases. For most of European history there was a considerable gap between the common everyday language people spoke and the language of literature, a gap that eventually meant the modern Romance languages (French, Spanish, Italian, and the like) were more or less mutually unintelligible for average people, and thus all the literature written in Latin were inaccessible to them. Considering that only after the invention of the printing press did regionalized texts become widely-distributed, and that Latin continued to be the preferred language for many academic journals well into the 19th century, this could not be avoided.

Ancient graffiti, like the many samples found at Pompeii (1st century AD), tells us that the average Roman probably spoke a language very similar to written Classical Latin, but there are still some simplifications and mistakes in the specimens that suggest there was a discrepancy between how they spoke and what a modern Latinist typically achieves. As for the middle ages, it somewhat depends on the century; for a while, spoken vernacular was identified as "vulgar" (meaning "of the people") Latin rather than its own language, and educated scholars from the provinces simply corrected verbalized errors in their writing. By the time of the Renaissance, the Romance languages were not only distinct, but so far removed that Dante Alighieri (of Divine Comedy fame) was able to make the "discovery" that they were related, an event which could be called the "start" of historical linguistics in Europe.

It might be noted that rhetoric and logic didn't quite have the modern definitions either. We think of rhetoric (rather negatively) as specifically the skill of making convincing statements, most of what you learn in English class in high school (how to write an essay, etc.) also falls under the umbrella of classical rhetoric. Education in logic obviously did not contain any of the modern developments of logic—George Boole hadn't been born yet—but focused on translating arguments into discrete steps so they could be analysed for their truthfulness, often by reviewing famous speeches or dialogues (a popular ancient format of philosophical text). Because of this emphasis on studying works by past masters, which was also done in rhetoric, students tended to learn a lot of other philosophy while perfecting the trivium, not just rhetoric and logic—topics we'd now call metaphysics, theology, ethics, aesthetics, political philosophy, and epistemology. By the end of the course, the student would be prepared how to think, at least by the standards of the day.