This is a really broad questions, since the type of school, and method of teaching vary a huge amount, both in our world, and in the worlds of Star Trek. Are you talking private schools, public schools, religious schools, military academies, etc. Are we comparing them against primary schools, Starfleet Academy?

For example, we know in DS9, and TNG that humans have some similar-ish kind of school system to what we have now, but the curriculum is tweaked such that children learn what would be considered difficult topics for us, quite early on. Keiko runs a school on DS9, and in TNG, we both see a classroom, and there is a child who complains about having to take Calculus at one point.

In the 2009 movie, Vulcans learn through some sophisticated hologram/computer system, or at least, what they have learned is tested that way.

The Borg and Binar, by virtue of what they are, don’t learn so much as have knowledge and information directly uploaded into them.

The Betazoids might do something similar, teaching students via telepathy.

What do they learn in star trek schools

Generally speaking, they likely learn a lot of the same subjects we do, but tweaked for their various homeworlds, and/or the Federation at large. The curriculum might deviate a bit in science, since they almost certainly factor in subspace, and some basic information on temporal mechanics.

Schools in star trek are no longer about the transfer of knowledge.

They are about helping students grow their base of knowledge and of experience, which must start from within.

Help students grow their own understanding instead of handing one to them.

I think this would be wise:
https://www.quora.com/What-are-some-things-schools-should-teach-but-dont/answer/Harri-K-Hiltunen