Education has existed long before government run schools. A lot of schools were started by companies that provided grants to open schools to create a more educated working base.
In a free market, you would have genuine competition, companies, businesses, actually competing to teach your child. Some might offer a free ipad, some free lunches, some might offer free pick up and drop offs, some might just offer a stellar education, but you would get ultimate quality over time, as the schools that fail children will go out of business and the schools that produce smart and successful children will prosper. You'd also have more choices, you would have schools for bad kids or mentally ill, schools genius kids, small schools, large schools, technology schools, agricultural schools, schools focused on the arts, cheap online schools for home training, etc etc. In all honesty, the educational system would be much much much better.
Of course, the first question is going to be "what about the poor?" The very fact that most people ask the question should tell you that most people are generous and caring, and will do what is possible to get education offered to lower income people. It doesn't take 1 million dollars to teach a child. It just takes a person that cares about teaching kids to make it happen. I often tell people about Marva Collins . She was a black teacher in inner city Chicago ,who taught at a public school that was failing the kids. She became disillusioned with government schools...she quit her job, turned her house into a school, and approached all the parents of the kids that the public schools have deemed "unteachable" and offered to teach them for a small fee. She worked with the poor parents who couldn't afford much, and did what she could. Those kids she taught wound up having better test scores than the government school children. Of course the government hated this, and tried to shut her down with stupid zoning laws and shit like that, but she eventually opened up her own large school that produced hundreds of educated kids that come from poor communities. She didn't have money to do this, but using simple free market economic philosophies, she was able to make it happen. Now imagine if this happened on a larger scale, without all the taxes, without all the government red tape and bureaucracy? Man, it would be unbelievable. Education would also be cheaper simply due to competition and less overhead to support a bloated bureaucracy of administrators, teachers unions, and stupid laws.
Do some research on it bro, their arguments about social good are easily dismantled.