Public schools and private schools
Most of the great 'public' schools of England took boarders and charged fees. They were for the sons of the rich, and were not public at all. They taught their pupils Latin and Greek, but not much mathematics, and no science, history, or French. Rich men's daughters stayed at home. They were taught by their mothers, or perhaps a governess.
Dissenters (Protestants who did not belong to the Church of England) had their own schools. They were the best in England. Only they taught science, commerce, French, and German. Many of the businessmen and inventors of the Industrial Revolution had been to dissenters' schools.
The very poor did not send their children to school, because they could not afford the fees. Also, they needed the wages that their children earned. But men and women who had just a little money sent their sons and daughters to private schools. Some of these were really awful. They had no proper books, desks, or classrooms, and the teachers were ignorant, greedy, and cruel.
Charities ran schools for the poor in some towns, but there were not enough of them. Then, in 1780, Robert Raikes opened his first Sunday school in Gloucester. His aim was to occupy the children of the poor, who worked in factories on week-days. Some of the children learned to read and write. All of them were taught to obey 'their betters'.
Walter Robson: Britain 1750 – 1900; Oxford University Press, 1993/2002, page 79