Roman education in the Republic of Rome (Around 750BC to 300BC) was practically non-existent. The children would be taught what they know by their parents, they would have been taught the basic skills they required to live (E.g. Domestic, agricultural and military skills. They would have also been taught how to behave and how to live up to their responsibilities as a Roman citizen).
The first schools in Rome would have appeared in the 4th century BC, this was a result of the plebeian class being allowed to get involved with the politics of the country. These schools were also mainly concerned with giving children the knowledge of how to behave in society, and the skills needed to live.
At the height of the Roman Republic formal schools were starting to be assembled. However they wouldn't have been free to attend and they were targeted at the middle to higher classes of society. Both boys and girls could have received an education, but it is likely that they would have been educated separately.
Following the conquest of Greece in around 146BC the Roman education system faced reform. As the Greeks (Especially Athens) had already refined their educational system the Romans adopted similar principals. Many Greeks were taken as slaves to Rome to be used as teachers for the richer Roman children. However, there were differences in what was taught at these schools, in Greece they focused on music and athletics whereas the Romans did not see either of these important. (The Romans only saw athletics as a means to keep their army fit) This new Roman education system had a similar structure to modern day schools; they would attend a first school and then progress to a secondary school and so on. The progression from these schools was based more upon ability rather than age.
Children of middle to lower classes would have attended a first school, and they would have been taught by a 'litterator', the Roman equivalent of a teacher. This was seen as a respectable position within society, however would not have received a great wage. They could have been found in many places across Rome, from a private residence to a gymnasium!
At around the age of ten a child would leave their litterator and depending on their gender and how wealthy they were they would take different paths. Girls at this age would stop education and would focus on finding a suitable husband. Boys of lower classes would stop their education in order to pursue work to help feed their family. However the boys of upper classes would continue their education at a higher level. They would now be taught by a 'Grammaticus' who would help them develop their writing, speaking, poetic analysis skills, they would also be taught how to speak Greek.
If after this stage they wanted to continue their educational career they would be taught by a 'Rhetor'. However, few went on to this stage as only a few trades required such an extensive education (E.g. lawyer, politician). At this stage they would be taught public speaking, geography, music, philosophy, literature, mythology and geometry.