Also, did the parents have to pay for their kids to attend?
When did the individual states start involving themselves in primary and secondary education?
Presently, the states are very intertwined with the local education systems. Generally, I am interested in how the education system in America evolved to its present state.
Remember that each state has a different setup. In some, schools are organized by township ("town" in New England), while most states have school districts that are independent, both politically and geographically, of any other unit of government. I'm not aware of any US state that administers schools at the state level.
States have become more involved in funding in recent years, but this varies dramatically from state to state, and have also become more involved in testing/accrediting/regulating local school districts.
For the Great Plains, the most common arrangement was for schools to be organized by townships (a subdivision of a county generally 36 square miles in size). The government land sales of that part of the state would have reserved Section 16 (one square mile) of land for the support of the township's schools, and the township school board would sell or lease that land—and sometimes use part of it as the location for the township's schools. In more arid parts of the West, schools were more often organized by the counties or large parts of them, in independent districts.