That isn't quite right. Schools are in the same category as fire and police departments and hospitals. The government pays the school districts to perform a service.
The reason I think it is important to understand the difference is that when a school does very poorly, the government can intervene and take over operations. This is considered about the worst thing that can happen to a school. It was part of the No Child Left Behind act.
It has also proven ineffective, as the schools taken over have yet to show any improvement and struggle to attract employees.
All of those are the government, including hospitals if they're public. There are different levels of government, and the examples you gave are generally at the local level.
As far as First Amendment rights of students, they exist, but not to the same extent as for others. The reason being a) These are minors b) School is compulsory, so all the students are required to be there. So the school can typically regulate threatening or hostile material, but not material that is merely politically offensive.