There may be no mention of schools in The Shire, and we may have Sam being taught 'his letters' by Bilbo, but this does not add up to the population being mostly illiterate. On the contrary, there is plenty of evidence that the general population of The Shire could read and write.
Bethberry has already mentioned calendars - if the population was mostly illiterate how would they be able to read calendars? There is also a Message Service which would not exist were most of the people unable to read and write letters. The Shire also has a legal system of sorts, having both Shirriffs and Bounders, and Lawyers dealing with property and inheritance law. Presumably the Mathom House also has those little cards telling visitors what the various exhibits are?
There's also economic evidence in that most people seem to have leisure time, suggesting they have gone beyond the limits of feudalism and have entered a more modern age where there is commerce (so Bilbo could go and buy a new weskit instead of having to have a wife who could sew him one) instead of everyone just having to sort themselves out, and they have specialised labour (Sam being a gardener for example, not just a farm labourer).
Yes, there are no schools we know of, but that doesn't mean there is no education! There may even be
more education