Your children become eligible to go to school when they reach four years old, and then for apprenticeships at eight years old. You can gain control of your children as playable characters when they reach 16 years old.
For 500 gold pieces, you can send one of your children to the monastery school for basic education. For 1,500 gold pieces, you can send them to the guild school where they can gain basic education and guild fame. For 5,500 gold pieces, you can send them to the school of the nobleman's mansion, where they will gain imperial fame (note that they will not gain guild fame with this choice).
Your children can apprentice into different professions, becoming patrons, craftsmen, scholars, or rogues. All of the options for apprenticeship cost 1,000 gold pieces. Note that your children will gain two levels if they apprentice into the profession they started off with rather than changing professions.