In reality, almost nobody had very good educational opportunities. Children of slaves (often more than half the population) and lower class workers (another big fraction) seldom got any education at all: just enough training to do their jobs. And only in very unusual circumstances were girls educated beyond the "womanly virtues" of spinning, weaving, cooking, making babies (or not). Some upper class girls were taught in the home or sent to elementary schools to learn reading or writing, but they weren't allowed to pursue higher studies with mathematicians, philosophers or rhetoricians. Most folks, of course, considered education for girls to be a waste of time and money.
So there was a small educated upper-class male minority. In the early days of Rome, boys were taught by their parents, and this pattern persisted longer in less wealthy families than among the really rich. If money was available, a rich father would hire a teacher (or buy a educated slave, usually Greek) who would manage the education of his male offspring. Some boys would be sent to private schools. In some bigger cities, "public" schools were actually established, but "public" meant that they were government funded, not that they were available for anyone who wanted to go. They were, in fact, only for male offspring of patrician families that were temporarily short of funds or for male students, of whatever class, who might have upper class sponsorship.
A Roman school, whether private of public, would often be nothing more than a one-man operation in a single room or even in a shop booth in the marketplace. Teachers were overworked and underpaid, putting in long hours for the same wages as the least skilled artisans and manual laborers. There are recorded complaints that their income was often less than two thirds of what they needed to support a family. They didn't starve, however -- just went on the dole like any other underpaid workers.
In addition to reading and writing and simple math (use of an abacus), a boy's basic education would include law, morality, and physical training -- the last, of course, with a decidedly military slant. Teachers were expected to use corporal punishment (caning and flogging) not only for misbehavior but also for poor academic performance.
After the basics had been absorbed (or were beaten into the students), those who could afford to studied higher mathematics, philosophy, and rhetoric. The richest kids might even be sent abroad to Greek academies or Egyptian salons, and parents who couldn't fund such grand tours hired the best they could afford in Rome.
A well-educated Roman male was ready to begin the long march to the top of the Roman political-military structure. If from one of the top few families, and or if correct marital alliances could be forged, there was no reason why he should not aspire to the highest offices of the state. He also, of course, had to be successful at arms and a better than average political infighter, but that, after all, was what he was educated to be.
For more info on education in Rome, try these Internet sites:
For info on education about Rome, go to these sites:
http://atschool.eduweb.co.uk/nettsch/time/romans.html (for primary students) and