Yes, schooling was prevalent. One of the major projects the Bolsheviks undertook was education of the masses. The 1920s and 30s saw the beginnings of that project. However, the project was far from a top-down monolithic process. There were quite a few different ideas about what education should be in the Soviet Union and different groups clashed over them. Lenin, for example, believed strongly that education was about politics, where was Lunacharsky (Commissar of Enlightenment) took a broader view about education as a more general force for ending class conflict and promoting a cultured population. In both cases, education was very specifically and explicitly meant to reach the peasantry and the working class. A big part of the Bolshevik project was to create a "new Soviet man/woman" and that, by their conception, required a well educated population.

In early years after the revolution the Narkompros (The Commissariat of Enlightenment - that the above-mentioned Lunacharsky led), proposed a very labor-oriented reorganization of education. It focused less on individual subjects and more on general themes (labor, society, etc). Things were often very provisional and unclear. By the time NEP (the New Economic Policy that came at the end of the Civil War and which brought some elements of a market economy back) came around, funding dropped off substantially for schools.

Also, throughout the 1920s and 30s the Bolsheviks led a campaign for the elimination of illiteracy (remember, a good socialist needs to be able to read Marx!). This was especially aimed at the peasantry, which was the least educated group in the country. The Bolsheviks made a big push to send people out into the countryside to teach people to read. (This is, in my estimation, comparable to other Bolshevik infrastructure building projects, like the "electrification" of the country). The project was only modestly successful at first, but by the end of the 1930s literacy had dramatically increased.

That's to say nothing of the on the ground reality that teachers often balked at Bolshevik curricula. Parents too were less interested in this kind of education and wanted more traditional forms of education for their kids. In many ways this was the defining feature of early Soviet education. Yes, more people were getting educated, but teachers and parents alike fought for modes of education they wanted and the state, Narkompros in particular, was forced to negotiate the terms of education in the country as a result. Things became more stable, if more authoritarian, in the 1930s.

The classic work on this topic is Larry Holmes The Kremlin and the SchoolhouseBloomington: Indiana University Press, 1991.

You might also be interested in Sheila Fitzpatrick's work which deals a lot with educational and cultural policy in the 1920s and 30s. See her: The Commissar of Enlightenment and Education and Social Mobility in the Soviet Union 1921-1934.