If you wish to delve deeply into the Roman education system, you should probably start with the two main sources for Roman education, the writings on oratory by Cicero (De oratore [Loeb edition]) and Quintilian (Institutio oratoria [Loeb edition]). [Suggested by, for example, "The Legacy of Roman Education," by Nanette R. Pascal; The Classical Journal, Vol. 79, No. 4 (Apr. - May, 1984), pp. 351-355.] Going further, you can find picturesque insight into Roman educational practices in the works of many Latin and Greek writers, including the sharp-tongued, exaggerating Juvenal (especially, Satire 7):

150 Or do you teach rhetoric? O Vettius! what iron bowels must you have when your troop of scholars slays the cruel tyrant: when each in turn stands up, and repeats what he has just been conning in his seat, reciting the self-same things in the self-same verses! Served up again and again, the cabbage is the death of the unhappy master. What complexion should be put on the case; within what category it falls; what is the crucial point; what hits will be made on the other side - these are things which everyone wants to know, but for which no one is willing to pay. "Pay indeed? Why, what have I learnt?" asks the scholar. It is the teacher's fault, of course, that the Arcadian youth feels no flutter in his left breast when he dins his "dire Hannibal" into my unfortunate head on every sixth day of the week, whatever be the question which he is pondering: whether he should make straight for the city from the field of Cannae, or whether, after the rain and thunder, he should lead around his cohorts, all dripping after the storm. Name any sum you please and you shall have it: what would I give that the lad's father might listen to him as often as I do! So cry half-a-dozen or more of our sophists in one breath, entering upon real lawsuits of their own, abandoning "The Ravisher" and forgetting all about "The Poisoner" or "The wicked and thankless Husband," or the drugs that restore sight to the chronic blind.
171 And so, if my counsel goes for anything, I would advise the man who comes down from his rhetorical shade to fight for a sum that would buy a trumpery corn-ticket - for that's the most handsome fee he will ever get - to present himself with a discharge, and enter upon some other walk of life. If you ask what fees Chrysogonus and Pollio get for teaching music to the sons of our great men, you will tear up the Rhetoric of Theodorus.
_{~G.G. Ramsay translation of Satire 7 of Juvenal from the 1918 Loeb edition.}

For more on Juvenal and education, see: "Three Notes on Roman Education," by M. L. Clarke; Classical Philology, Vol. 63, No. 1 (Jan., 1968), pp. 42-44.

Suetonius also deserves a place on the list of those who wrote about the system of Roman education, as "The Roman School Teacher and His Reward," by Rodney P. Robinson; The Classical Weekly, Vol. 15, No. 8 (Dec. 5, 1921), pp. 57-61, shows:

"[Suetonius] wrote a book 'De Grammaticis et Rhetoribus', dealing with the lives of famous Roman teachers of grammar and rhetoric, thereby paying a tribute to the profession to which he himself seems to have belonged at one time."

Robinson also quotes Tacitus (Dialogus 29) on Roman education:

"'The children are turned over to the care of slaves who are incompetent for any serious service. Their parents do not train their little ones in goodness and self-control, but in wantonness and pertness, which gradually grow into insolence and disregard both of self and others....'"

Suetonius, Juvenal, and Tacitus are among those who are absent from the list of sources in "Historiography and Roman Education," by Michael Chiappetta; History of Education Journal, Vol. 4, No. 4 (Summer, 1953), pp. 149-156, but that doesn't invalidate the article's potential for leads. Below, I list Chiappetta's ancient sources on Roman education, as a starting point for your research. The names are chronological. My hyperlinks -- except for the first -- go to biographical information, rather than English translations of the pertinent material on Roman education, although the Internet History Sourcebook contains specific passages for Pliny, Horace, and Martial. The first item on the list goes to highlights from the early Roman law code.

1. Laws of the Twelve Tables
2. Terence
3. Cato the Censor
4. Polybius
5. Catullus
6. Cicero
7. Sallust
8. Varro
9. Nepos
10. Vergil
11. Dionysius of Halicarnassus [see Sources for Roman History]
12. Horace
13. Livy
14. Ovid
15. Vitruvius
16. Seneca the Elder
17. Persius
18. Seneca the Younger
19. Quintilian
20. Martial