Asalaamulaikum respected brother,
Insha'Allah, you may be able to help me answer these questions for a friend of mine.
What is Islam's view on the parents' (especially the father's) responsibility to educate their children? Additionally, what type or manner of education is most Islamically correct (i.e. private or public school etc.)?
Al-hamdu lillaah. Praise be to Allaah.
No doubt that a child's education is one of life's necessities, and as such spending to achieve it is obligatory on the parents. However, the level of education that a Muslim should provide for his children is not necessarily the highest level possible. This is because a child can lead a perfectly normal life with only, for example, high school education.
And as for whether private schools are better than public schools from the shari'ah (Islamic jurisprudence) standpoint, I do not think that there is a general rule by which we can favor one schooling system over the other. It all depends on the specifics of each individual case when assessed with respect to curricula, teachers' skills, and the school district credibility.
Wallaahu a'lam. And Allaah knows best.