I read in an excerpt of Paul Veyne's History of Private Life that education in early Roman Empire was not intended to make people ready to enter society (as we understand school today), but to educate them in the field of rhetoric (social prestige) and able in sports. The most educated Roman can still be illiterate and may have his slave to do the writing and reading.
With the influence of Christianity which seems to emphasize the importance of writing and reading, how was schooling and education in Christian Roman Empire? I.e. what was the purpose, what was taught, was literacy now a mark of educated mind like we expect from education today? Etc.
While I think your main question is still a valuable one (the influence of Christianity on schooling, education, etc.), your premises are misguided.
Rhetoric isn’t the art of social prestige, it was in the Roman context the field of studies that prepared you for public life, because so much of your success (in the upper classes), depending precisely on your ability to speak well and persuasively, and that is what rhetoric is. But we shouldn’t be mislead into thinking that that is ‘all’ that rhetoric was. It wasn’t just a short course in speech-writing. Rhetorical studies encompassed and shaped all of what we would now call literary-studies and literary-theory, grammar, linguistics, philology, poetics, and good doses of culture and history. Technically legal studies were thought of separately, but rhetoricians often worked in the realm of law and were familiar with it.
While the most educated Roman might have a slave read aloud to them and write for them, it is very unlikely that they would themselves be illiterate. Rhetorical education was education, and upper class Romans were educated and educated thoroughly.
To understand Rhetoric in the ancient world, the best starting place remains the works of George Kennedy on it. One could start with his “A New History of Classical Rhetoric”.