What is the responsibility of parents in education, and what is the quality of education required Islamically (private, public, etc.)?

Educating a son is one of life's necessities, and the father must spend to fulfill this necessity. Therefore, he must educate him, though not necessarily through all stages. As for the preference between private and public schools, there is no fixed rule in Sharia. The preference is determined by the curricula, the teaching staff, and the system followed.