Classical education has three phases, each with a different purpose. The phases are roughly coordinated with human development, and should be exactly coordinated with each student's development.
In the framework of classical education, primary education teaches students how to learn. Secondary education then teaches a conceptual framework that can hold all human knowledge (history), and then fills in basic facts and practices of the major skills (perhaps in a simplified form) of every major human activity. Tertiary education then prepares a person to pursue an educated profession, such as law, theology, medicine or science.
Primary education was often called the trivium, and covered grammar, logic and rhetoric.
Grammar consists of language skills such as reading, and the mechanics of writing. An important goal of grammar is to acquire as many words and concepts as possible. Very young students can learn these by rote.
Young adults can learn logic, the art of correct reasoning. Modern logical systems are remarkably easier to learn than classical logic.
Rhetoric, debate and composition (which is just written rhetoric) are taught to somewhat older students, who then have the concepts and logic to criticize their own work, and persuade others. Rhetoric is concerned with making communication clear, reasonable, and persuasive. Hopefully, a student has already learned to reason correctly by studying logic.
Ideally, logic and rhetoric should be taught by the Socratic method, in which the teacher raise questions, and the class discusses them.
Secondary education, classically the quadrivium or "four ways," classically taught astronomy, arithmetic, music and geometry.
In modern terms, these fields might be called natural science, accounting and business, fine arts (at least two, one to amuse companions, and another to decorate one's domicile), and military strategy and tactics, engineering, agronomy, and architecture.
These are taught in a matrix of history, recapitulating the natural development of each field for each phase of the trivium. That is, in a perfect classical education, the historical study is repeated three times: first to learn the grammar (the concepts and design techniques in the order developed), next time the logic (how these elements could be assembled), and finally the rhetoric, how to produce good, humanly useful and beautiful objects that satisfy the grammar and logic of the field.
By the time a student has completed a project in each major field of human effort, they often have an excellent idea of what type of profession they would like to pursue.
History is the unifying conceptual framework, because history is the study of everything that has occurred before the present. A skillful teacher also uses the historical context to show how each stage of development naturally poses questions and then how advances answer them, helping to understand human motives and activity in each field. The question-answer approach is called the "dialectic method," and permits history to be taught Socratically as well.
The Socratic method is the only known technique to teach people to think correctly and critically for themselves. In-class discussion and critiques are essential in order for students to recognize and internalize critical thinking techniques.
The early biographies of nobles probably show the ultimate form of classical education: A tutor. Other forms of instruction seem to be attempts to achieve similar results for less expense.
A classically educated person is intensely skilled, highly disciplined, broadly educated, and if taught Socratically, an amazingly supple and accurate logician and rhetorician.
Accurate information about classical education is difficult to find. People took it for granted for generations, and then within one generation, Deweyism became the established method for public schooling in the U.S, and classical eduation became rare.
The best available modern reference is "The well-trained mind, classical education at home," by Jessie Wise and Susan Bauer.
For a history of alternatives, see education reform.