Would education be free? It's never mention one way or the other on the show. Starfleet Academy training would come in exchange for agreeing to time in service. Wesley's ship board education, and Jake's home schooling would come from being dependants of Starfleet personnel. Part of the contract.
Keiko briefly opperated a school on DS9, but any financial arrangements were unseen.
The ability to achieve Star Trek economics is, of course, debatable. However, it would seem that the costs of any organization or enterprise, including schools and universities, is almost negligible. Quark's business had a lease with the Federation, but it collected neither the rent not the power bill. In many countries, the cost of a university education is already very low for citizens. It is feasible that higher education would be free in this universe.
Well they still have jazz.
They eradicated good
Jazz and left the awful Smooth Jazz behind.