Parents. You make a good point in that almost any developed civilization has schools, but I think the Elder Scrolls operates on a parallel to the middle ages where you simply learned what you needed for your future job, which was often within the family (I am assuming).
A farmhand may learn very basic reading and writing, only used for communication and possibly book keeping for the farm. However, a scholar/scientist will likely delve deep into reading and writing for their line of work, recording their studies.
tl;dr it's quite easy for parents to teach their children the basics (or even more). my brother is doing it right now with his kids and its amazing to watch