How did the education system in Colonial North America work?

(There are quite a number of sources on the internet concerning colonial education, some of which are provided at the end for those who want more than the 'highlights' below.)

The education available to the colonists depended very much on two factors: where they lived and how wealthy or what class they were. Private education was all there was initially and included

home, school, church, voluntary associations such as library companies and philosophical societies, circulating libraries, apprenticeships, and private study.

The wealth and class side is quite well summarized by Colonial Education

Colonial Education of the Upper Classes - The boys from upper class families were taught be private home tutors and then sent to college or university. Many of the Upper Classes sent their boys abroad to English educational institutions in order to receive a university or college education.

Colonial Education of the Middle Classes - Boys from the middle class, the sons of merchants, ministers, doctors and lawyers, attended dame schools, elementary schools and grammar schools. Only occasionally would they attend college

Colonial Education of the Lower Classes & Indentured Servants - Limited colonial education - apprenticeships were available to some lower class boys

Slaves - Slaves had no education and in the Southern colonies slaves were forbidden by law to learn how to read and write

For many colonists, as stated by Robert A. Peterson,

Education in early America began in the home at the mother’s knee, and often ended in the cornfield or barn by the father’s side. The task of teaching reading usually fell to the mother

Darrell J. Kozlowski in Key Concepts in American History: Colonialism observes that

As the English colonies became more settled, education grew in importance. In New England, close relations between church and state led to the rise of a public school system designed to teach reading and religion. For example, the Massachusetts Public School Law of 1647 required each each town of 50 families or more to support an elementary school.

In what was then New Amsterdam, Peter Stuyvesant helped found public schools so that

By 1661, New Amsterdam had one grammar school and two free elementary schools, as well as 28 licensed teachers.

Source: Kozlowski

In some areas, it took much longer for schools to be founded. The first pioneer settlement was not established in Kentucky until 1774, but a school was not long in coming as

Jane Coomes, who came from Maryland with her husband William, started a dame school at Fort Harrod in 1775 or 1776.

Progressing to college level (which, as noted in the question could mean boys as young as 15), Harvard led the way:

Nine young men made up the first cohort of graduates in America, the Harvard Class of 1642. These first American bachelors of arts were inheritors of the “Oxbridge” academic tradition of England, yet they were also products of very colonial American concerns.

Source: William N. Haarlow, Great Books, Honors Programs, and Hidden Origins

Haarlow also says that

The early curriculum of the colonial colleges was thoroughly classical and prescribed. Emphasis was on the three Aristotelian philosophies: natural, moral, and mental (that is, physics, ethics, and metaphysics), ancient languages (Greek, Latin, and Hebrew), and especially divinity.

Unsurprisingly, educational opportunities for African Americans in the Antebellum era were 'practically nonexistent' (Haarlow) while for (white) women it

was similar in numerous respects to male education at the time, but it also introduced for the first time some of the subjects that would later be found in most colleges and universities. Women’s colleges taught Latin and sometimes Greek; foreign languages, with an emphasis on French; mathematics; and science, which demonstrated God’s designs. They also pioneered the study of the fine arts, particularly music and art, which along with embroidery, dance, and conversation, made up the “ornamental” subjects in the curriculum.

Additional sources:

• Education in the Thirteen Colonies (Wikipedia)

• History of education in the United States: Colonial era (Wikipedia)

• A History of Education in Kentucky by William E. Ellis

• Education in Colonial America (Study.com)

• Schooling, Education, and Literacy, in Colonial America