Here is a similar thread from some time ago with links to several others: https://www.reddit.com/r/AskHistorians/comments/1qhue3/im_a_student_at_oxford_university_in_the_15th/

I don't have sources off the top of my head, but I love medieval France so here goes.

High Middle Ages. France. For hundreds of years the monasteries have been the centers of learning. That might be a loose usage of "learning", they were the only place for education (for men), but they weren't dedicated to liberal learning, but studying scripture and the writings of Church Fathers (e.g. Augustine or Jerome). By the turn of the twelfth century cathedral schools began to compete with the monasteries as centers of learning. The general focus began as a similar goal: learn about the doctrines of the church. But there were certain differences, most notably that many teachers and students at cathedral schools did not take vows. By the end of the twelfth century these schools became the first universities. Monks hated that, but that is beyond the point.

I suggest looking up Peter Abelard. He started in the cathedral schools and became a monk. His Sic et Non is very interesting. I'll spare myself gushing about this work and let you look it up yourself (unless you want me to gush).

As for curriculum, it was very different. The core of education was the trivium: logic, grammar, and rhetoric. Once those were complete, a liberal education was completed with the quadrivium: arithmetic, astronomy, music, and geometry. Once you were educated in those, if you wanted more education you learned about the robed professions: law, education, or the church. Notice how judges, professors, and priests all still wear robes: these were the most educated individuals in medieval Europe.

So, long story short, church schools were secular in comparison to the monasteries, students flocked to them, and as they grew they became independent of the church.

Look up the twelfth century renaissance. Many historians consider this more important than the Renaissance. I can go into why twelfth century northern France is so important if there is interest.

Sorry for not sourcing this, I'm at a bar and on mobile right now. If you need sources I can provide tomorrow. Also I feel this answer was short, feel free to ask for more details.

There is sparse information about it, but Nalanda University is thought to have been set up some time in the 6th Century BC. Unfortunately it was destroyed by Khilji in his raids in the 13th Century. It's size causes many historians to label it a University rather as opposed to the more informal education provided elsewhere at this time. Moving into folklore now, but it is believed that Buddha himself taught here at one point.