This answer will draw very heavily from Hannam‘s „God’s Philosophers“ as the book fits the question pretty neatly.

First, just like today, you made your bachelor degree. For a bachelor of arts, you had to study 3-4 years and your studies would include grammar (advanced Latin, you were supposed to know Latin already, as it was the prime language at universities), dialectic (logic, mostly based on Aristotle) and rhetoric (what it says, including argumentation). These three subjects were so fundamental, that their name, trivium, christened the word “trivial”. During the lectures, the lecturer would read out of a book (the students didn’t have any, as they were too expensive) while the students made notes. In German the word “Vorlesung” (lecture, literally “reading”) still reflects this. Once you had mastered the trivium, you would move on and be assessed during the “determination”, after which you received your degree as a bachelor of arts.

Then you would move on to more specialized topics. For a master of arts, you studied another few years and covered four more “liberal arts”, mathematics (The abacus and Arab mathematics), geometry (using Euclid’s Elements), music and astronomy (using Sacrobosco’s “The Sphere”, which alone should tell anyone that virtualy all Medieval scholars believed to universe to be spherical). In addition, you studied ethics, natural philosophy (proto-science, meaning even more Aristotle, or rather a commentary on Aristotle as his originals are not very approachable) and metaphysics.

After making your master’s, you were finally allowed to study the Queen of all Sciences, Theology.

Of course, studies in medicine and law would be different, e.g. a medicine lecture at the university of Padua might have lectures where a helper performs an autopsy while the professor explains the organs, and cover books by Galen or his Arabic successors, like Avicenna.

It should also be noted that this was very expensive. Now, some rich people might send their children to university, so they don’t need to worry. Others might try to finance themselves by working as tutors. But many aspiring scholars found themselves unable to foot the bill. In this case, very good students could hope to join a religious order as monks, as these would then finance their studies in order to get a more skilled cleric in the bargain (e.g. the Dominicans pursued the objective of out-debating heretics, so for them logic and rhetoric were very important). More affluent students, however, were often infamous for their hedonistic lifestyle.

If you are looking for actual STEM, the natural philosophy part in the arts degree is as STEMy as you will get at a Medieval university. To their credit, the Greeks and Romans weren’t much better in this regard and they in a way inherited their distaste of philosophers doing manual labor. Otherwise, it was mostly philosophy, law and medicine.