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.
I had always considered Gene Roddenberry's vision of the future in TOS and TNG to be a rare utopian one.
I'd consider it a sad thing if people still were to pay for education and in general if the size of your wallet were to determine what you can or cannot do.
Ideally, in a positive futuristic society the skills and preferences an individual has should determine the career and not money.