As you said, by the 17th century many people were literate and had a copy of the Bible. The following figure shows the literacy rate throughout the last 500 years in several countries:
An interesting point for me is the sharp increase in the UK and Netherlands, both Protestant countries, compared to Italy and France which remained Catholic (Italy in particular as birthplace of the Renaissance is disappointingly low).
During the Middle Ages education was religious and few people cared for it but it was offered. From the Wikipedia page on the history of education:
Modern systems of education in Europe derive their origins from the schools of the High Middle Ages. Most schools during this era were founded upon religious principles with the primary purpose of training the clergy. Many of the earliest universities, such as the University of Paris founded in 1160, had a Christian basis. In addition to this, a number of secular universities existed, such as the University of Bologna, founded in 1088. Free education for the poor was officially mandated by the Church in 1179 when it decreed that every cathedral must assign a master to teach boys too poor to pay the regular fee; parishes and monasteries also established free schools teaching at least basic literary skills.
This shows that before the Reformation there was indeed an effort by the Church to educate people.
After the Reformation, the need to be able to read the Bible was increased due to the nature of the new doctrines. They emphasized study of the Bible as necessary for the faithful creating therefore the need for more education. But what were the tools that tried to cover the increased need? Again religious institutions aided occasionally by the state. From the same article:
In northern Europe this clerical education was largely superseded by forms of elementary schooling following the Reformation. In Scotland, for instance, the national Church of Scotland set out a programme for spiritual reform in January 1561 setting the principle of a school teacher for every parish church and free education for the poor. This was provided for by an Act of the Parliament of Scotland, passed in 1633, which introduced a tax to pay for this programme. Although few countries of the period had such extensive systems of education, the period between the 16th and 18th centuries saw education become significantly more widespread.
This shows that during and after the Reformation, efforts were increased both by the responsible authorities (Church, State, Charities) and by the commoners to achieve a level of literacy. Their goal was, as indicated here:
to create good
Christians, who would also be loyal and dutiful subjects.
In the same source is documented that residents of the Plymouth colony taught their children at first when they didn't have actual teachers. As knowledge becomes common ground for more, it makes sense to help spread it.
William Bradford, in the first years parents taught their
children themselves, the colonists having neither a suitable
teacher available nor the money to support one. By 1633,
that apparently changed, as least for young children.
So overall, the literacy level of people rose through efforts of religious and secular authorities with the combined increased interest of the peasants/commoners to actually learn.