The party's manifesto set out costly plans to reintroduce maintenance grants for students and get rid of tuition fees, which were increased to more than £9,000 a year under the coalition government.
Labour would overhaul the existing childcare system and extend the current 30 hours of free childcare to all two-year-olds.
On schools, the party pledged to limit class sizes to 30 pupils for all five, six, and seven-year-olds and to bring in free school meals for all pupils, using money raised by removing VAT exemption on private schools.
It would abandon plans to reintroduce baseline assessments and review SATs for Key Stage 1 and 2 pupils to avoid "teaching by test".
The manifesto also says Labour would not "waste money on inefficient free schools", while dismissing grammar schools as a "Conservative vanity project".
A pay cap for public sector workers would be scrapped to tackle a recruitment crisis among teachers, who will be given more input in the curriculum.
Improving skills was another key manifesto focus, with a pledge for free further education and the return of the Education Maintenance Allowance (EMA) to support young people on lower and middle incomes.
It promised to devolve responsibility for skills "wherever there is an appetite", giving more powers to city regions and different administrations.
The manifesto says: "When the 1945 Labour government established the NHS, it created one of the central institutions of fairness of the 20th century.
"The NES will do the same for the 21st, giving people confidence and hope by making education a right, not a privilege, and building bridges where the Conservatives build barriers."
Russell Hobby, general secretary of the National Association of Head Teachers (NAHT), welcomed Labour's efforts to tackle "unprecedented cuts" to school budgets.
He said: "We know that the school funding crisis is hitting all schools. Committing to reduce class sizes to under 30 for all five, six and seven-year-olds is welcome.
"Rising class sizes is a consequence of real terms funding cuts, and one that school leaders are being forced to make."
"This General Election is a make or break moment for education.
"We're seeing unprecedented cuts to school budgets across the country and parents are beginning to realise the negative impact this is having on the quality of their children's schools."