I realise that education was reserved for the children of elites, so most people got no formal education at all. But for the ones who did, did it start at the same school age we have today or did they start later into their teen years.
And to narrow it down a bit, let's only consider Egypt, the Hellenic world, Rome and China.
I'll defer to experts on those various societies but can offer the answer to your question is highly contextual, based on a particular child's development, his (or possibly her) parents' intentions for their child's future, and social norms. First some quick modern era context and then I can speak to China and the Hellenic world.
The first thing is to shake off the modern notion that there's a link between stages of education and the exact number of times a human being has been around the sun. The use of a specific age as a signifier for when education begins (and ends) is a relatively modern phenomenon. Generally speaking, it's linked to the systemization of formal, publicly-funded schooling. The idea that a child would start school because they've spent a certain amount of time on the planet didn't become the norm until well into the 20th century. Prior to that, the decision was generally made based on variables including a child's ability to see to their own bodily functions, the ability to control their body and sit still, and a general ability to communicate with an adult. There are exceptions but it has long been seen as outside the purview of a teacher/tutor to see to a children's bodily needs, especially when that teacher is a man. The notion of expelling a child is also a modern framing. If a child couldn't do what the teacher/tutor expected, corporal punishment was an option or the teacher would ask the child's family to remove them (or the tutor would quit.) Until the feminization of the teaching force in the mid-1840, teachers/tutors, especially those hired by elite families were men. A child, especially one from a family with means, who was as yet unable to do what was required for his (maybe her) instruction, was likely to remain with a female care-provider, such as a nursemaid, nanny, or governess who might include basic literary skills such as route counting and reading.
Education and childhood historians, such as Chudacoff, use the term "weak age norms" to describe how humans treated the quantity of trips around the sun until the modern era. Imperial China provides an interesting example of what this looks like. Elman, in Civil examinations and meritocracy in late Imperial China (2013), detailed that men from the age of "fifteen to fifty" would line up to the take the exam. In addition, he stresses that the Chinese system for documenting someone's age followed a convention: The Chinese usually added one additional year of age during the first year, if a child was born just before the lunar new year. To reflect this double counting during the first year, I have used the Chinese term of sui for “years” in this volume.
Throughout the book, he identifies the age when someone took the test, but because of how ages were documented, that person may or may not be that exact age. The spread of ages of men speak to these weak age norms. Lü Liuliang, a physician in the 1600's, was writing a particular type of structured essay at age 8, indicating he had several years of formal schooling under his belt. However, he was never able to pass the provincial exam, suggesting his formal education was incomplete or lacking key lessons. Other examples from the book include Dong Qichang (1555–1636) who took and passed his first examination at age seventeen. Meanwhile, Li Zhi took his first exam in 1552 at twenty-five. While in the Ming period, the oldest test taker was in their 40's, by the Qing period in the 19th century, there were test-takers in their 50's and 60's taking and passing the examinations. Elman establishes this is a function of the examinations getting longer and more complicated, so if we were to try to work backwards to figure out the start of test takers formal education, we'd likely reach a misleading conclusion. Finally, the notion of "capping" should disavow us of the notion things happened at a certain time because of a student's age.
Frequently the rites of passage from childhood to young adult in wealthy families were measured by the number of classical texts that a child mastered at a particular age. “Capping” of a young boy between the ages of sixteen and twenty-one, for example, implied that he had mastered all of the Four Books and one of the Five Classics, the minimum requirement before 1787 to compete in local civil service examinations.
This concept of mastery learning can also be seen in Hellenic world. Boys were generally separated from their family, from younger boys, and from older boys based their level of development. While some writing speaks to a boy's ninth year as instrumental, others highlight his 7th year. Multiple texts talk about a young boy or man mastering a set of skills, concepts, or ideas before moving on to the next stage of his education.
As I said, I'll defer to scholars who are more familiar with the various societies, but from from a history of education perspective, it's fairly safe to conclude children (by which we mean the sons of the elite) in all of the places you mentioned started formal education when it made sense for them, sometime within the later half of their first decade of life.
Bloomer, W. M. (Ed.). (2015). A companion to ancient education (Vol. 120). John Wiley & Sons.
Chudacoff, H. P. (1992). How old are you?: Age consciousness in American culture. Princeton University Press.
Elman, B. A. (2013). Civil examinations and meritocracy in late imperial China. Harvard University Press.
/u/UrAccountabilibuddy has already given a great answer which is actually pretty consistent with what we find in Graeco-Roman Egypt, and likely to some extent the Graeco-Roman Mediterranean by extension. I am adapting this from an older answer of mine.
Some of the most important pieces of information about educational traditions in Graeco-Roman Egypt are the literary accounts of Hellenistic pedagogy, and archaeological finds of school materials. For example, school exercises or teaching materials were often written on less degradeable materials like flakes of limestone, discarded potsherds, or tablets. These include Hermeneumata, which were essentially school handbooks, and other pedagogical tools like writing exercises.
Although this answer does focus on Egypt, many points hold true for other parts of the post-Classical Mediterranean, and I will try to distinguish between Graeco-Roman Egyptian and broader Hellenistic/Roman trends.
Formal education in the Hellenistic world had no enforced academic standards or accreditations that ensured uniformity of quality and curriculum. As such, there was no set standard age at which individuals were expected to have achieved some educational qualifications. The educational goals and methods of the classroom were under the control of the teacher who in turn, had to have the trust and cooperation of the parents. However, there were long standing pedagogical traditions which dictated the standards and ideals of teachers. This included the transmission of a canon of literature and knowledge which had become established as the basis of a cultured individual’s education, as well as average ages at which elite youths began to be educated in certain areas.
Unlike a modern education which is aimed at providing students with a basic level of aptitude in the sciences and mathematics, Greek education was mainly concerned with more theoretical, literary, and rhetorical forms of education. In Classical Greece, philosophy was considered another essential part of a proper education, but it seems to have lost a degree of its status as in the classroom by the Hellenistic period. This meant that a student might hope to achieve a high degree of mastery in the area of rhetoric and argument, and a comprehensive familiarity with the great works of literature, but they would have a relatively basic level of numeracy and probably virtually no other tutoring in technical fields unrelated to their vocation.
If a youth did study a "science" in any depth (such the body, mathemetical theory, astronomy/astrology, or alchemy) in any depth, it was usually because they were training to be a physician, mathematician, astronomer, alchemist etc.
For those children and adolescents lucky enough to receive a more formal education, this typically meant being part of a large study group which met regularly to study under a tutor. Raffaella Criboire in Gymnastics of the Mind cautions against imagining the ancient school as more orderly and modern than it actually was. Criboire compares the unstandardized, often mixed classroom of the ancient Mediterranean to the one-room schoolhouses of the 19th Century US in their variable quality and effectiveness. In the Ptolemaic period, these classrooms were usually not found in dedicated school buildings but were set up by teachers on temple grounds, private homes, and often enough in the open air (even street corners!). School buildings are more well attested to in the Roman period, but this mobile classroom was still probably the more common mode of teaching.
The education of the uppermost echelons of the elite was much more private and focused than what I just described. For the more elite circles of the aristocracy, education involved personal tutors in the home. This was deemed necessary because of the need to train children for their eventual roles as political and military leaders, and to provide them with the cultured background which was expected of the civilized elite. Some older students attended the classes and discussions of famous teachers in Alexandria which was the capital of Greek intellectualism for much of the Ptolemaic period.
The first instructor of any kind that elite children would have had were their pedagogues, just like other elite men and women. These pedagogues taught and cared for young children of the elite, reading to them and teaching them basic knowledge. However, it should be stressed that these individuals were not professional teachers in the way we might imagine. They are better imagined as general caretakers who looked after these children and taught them basic skills along the way. As children grew older, they eventually outgrew their pedagogues and became slightly more independent with each milestone they hit in Graeco-Roman culture. It was at this time that they moved onto more formal educations.
The symbols of the Hellenistic teacher, their chair and rod, demonstrate the relationship between pupil and instructor. Various scenes from the Mediterranean portray male or female teachers sitting and working while their pupils either stand before them, recite, or present them with their work. The rod was for exactly what you might have assumed as corporal punishment was readily employed by teachers as a means to discipline their students.
Alongside, grammar and rhetoric, boys and girls studied music and dancing. The playing of instruments as well as the recitation of songs and lyric poetry was an important social event, and youths participated in competitions as well as socializing through the practice of music. Dancing and athleticism also went hand in hand, as the Greeks idealised the flexible and lean body, with the smooth movements of dancing being a good form of exercise. Other athletic events, such as running, were participated in by boys and girls alike.
The gymnasium was another important centre of physical and intellectual education in the Ptolemaic period. These gymnasia were patronized by the elite, and were actually the focal point of urban Hellenistic city planning. It was also a mark of citizenship as only a citizen of the city could enroll, making enrolment a marker of status. Enrolment in the gymnasium was one of the most important stages in a young Greek's life, and typically occurred around age 14, as a step towards adulthood. This signalled the age when they were taxed as an adult. Participation in exercise and festivals at the gymnasium was an important part of a person's adult social life.
Hellenistic education was typically structured by three somewhat porous stages of learning:
• The primary level of education where students mastered literacy, this typically began around age 7 but there was a great degree of variance. It was not uncommon for students to begin later. Papyrological evidence has shown that it was at the primary level that the majority of female teachers taught, with female teachers at higher levels being more rare but also making a bigger impression in the historical record.
• Grammatical education where students studied poetry, and in the case of the Alexandrian aristocracy, this was taught by individuals who had experience composing poetry. This was taught sometime after students had attained a certain literacy, and typically before rhetoric. The famous names of Greek poetry such as Homer, Sappho, Pindar, Theognis, and Anacreon were studied, as were more contemporary authors such as Callimachus and Theocritus. This required students to accurately read, deconstruct, and critique poetry, and if this all seems very bourgeois, that was exactly the point.
• Rhetorical education, where students mastered oration and argument, generally began around age 14-15. This was not merely debate club, rhetoric was one of the essential components of a Greek education. The majority of students who studied at this level intended to pursue careers as politicians or lawyers, and as such this was necessary vocational training.
Children in Ptolemaic Egypt did not advance to all stages of education the way that children in school are expected to complete their education today. In fact, most students did not continue on to the secondary level, and fewer still continued on to achieve what was considered a complete education. The distinction between students’ level of mastery had little to do with merit, and was primarily dictated by their social background and their prospects. The advanced education provided to the aristocracy was a way to reinforce their own prestige and to provide them with the know-how to hold positions of power later in life.
It is important to note that even the most basic level of education was available only to a select few, the children of the elite. For girls especially, non-vocational education was accessible only to those who came from elite backgrounds. Girls were also less likely to study grammar because there were fewer incentives and pressures that might motivate their parents to provide them with that, and more pressures for them to marry at a young age and turn to the domestic sphere. Rhetoric was almost exclusively studied by a very small slice of aristocratic male youths who were going into the arena of politics or the law.
#Sources:
Gymnastics of the Mind: Greek Education in Hellenistic and Roman Egypt by Raffaella Criboire
The Birdcage of the Muses: Patronage of the Arts and Sciences at the Ptolemaic Imperial Court, 305-222 BCE by Rolf Strootman
Women in Hellenistic Egypt: From Alexander to Cleopatra by Sarah Pomeroy