Dames' schools served the working class, teaching the 3 R's usually abysmally. There were also small academies and individuals providing classes (see Robert Eadon Leader's "Sheffield in the Eighteenth Century" available from the Internet Archive)

Sheffield Grammar School - Hunters "Hallamshire" has a chapter on this (available from Internet Archive)
Free Writing School (est 1721)
The Boys' Charity School in East Parade (est 1707)