Firstly, i'm very new to posting on this forum (1st post) so if this is wrong please let me know so i can correct where necessary. I did a few searches looking for keywords regarding children and anything concerning the ideas i had and found nothing. I heard there is development for children but i would like to see a useful education system.
For the children:
A Nursery for children to go. Designate it similarly to a hospital. Tables and chairs will be needed. Perhaps using dyes and water to allow them to paint. Basic plastics could be made in the kitchen for them to play with. Once in a while they could go on excursions such as to your sculpture garden, to the well or the grand dining room. They could help by brining water to those in the hospital during sieges with a slight chance they drop the bucket on the way there.
They will learn the Learning skill, be kept out of the way and depending on the teacher can learn certain skills, if not entirely social skills.
Puzzleboxes, toys and some crafts optional.
For the children, i would recommend a School. this is a place to teach children skills and to keep them occupied. You could create a timetable where different dwarves of various skills take turns teaching the kids. There could be a military academy for children as well. Different teachers, such as a metalsmith, will teach predominantly their skills. But perhaps a farmer should still teach, upon occasion if he has a little experience in it, how to milk cattle.
The teachers should teach better the greater their skill in the subject. The children will learn better with greater learning skills (received from the nursery). Also, the greater the teaching skill of the teacher, the better the students learn. I like the idea of keeping your legendary weaponsmith away from the forges to help raise better weaponsmiths in the future. It could create a balancing act and long term planning.
Schools would require tables, a separate table for the teacher in the front of the class, chairs, graphite for pencils, paper, books, gypsum (which we usually call chalk in a classroom), a blackboard (made of metal). A direction should be set to face the chalkboard. Storage would also be necessary.
Mini forges, whatever their implementation will be, could be used to teach the smithying skills. Balls of yarn/threads and some dye for weaving, clothmaking and dying skills. Pots, carving knives, cauldrons for cooking, which allows for, in the general expectation of the game, for some fun.
Like the idea of a freak dying mistake happening, permanently staining a child vomit green. Or metalworking having tongs and a hammer soldered to his bones allowing him to only ever be a smithy.
If we implement education, we need tertiary level training.
A University is a place to learn skills not easily taught. A place for nobles to be taught their skills. An area to be taught, amongst other skills, the teaching skills which would be extremely valuable to send your high level dodger and commander to then teach the rest of the squad. This would also make your war veterans who cannot grasp anymore something very useful to do.
The University is also where social skills can be learned and practiced. It could have a debating hall where arguments and official debates take place similar to when a party is hosted in a meeting hall. Leveling up skills of the listeners and debaters (only those enrolled in the university may debate). This could create happy thoughts of differing degrees based on the level of the debaters. The more amusing the discussion and greater comedic level of the debaters, the more enjoyment. Perhaps some dwarves who are inclined towards knowledge prefer the more serious arguments, but let us leave such specifics to another time.
Temple: Increases the likelihood that your dwarves who spend time worshipping will be possessed and create an artifact. Most strongly for those who manage the temple (a monk or perhaps).
If, and this an idea i really like, the temple is organized concurrently as the university, we could have theologians for the temple and rational philosophers debate deities. What this could do is alter the relationships that dwarves have with their deities. We could create uneasy thoughts when the believers begin questioning and happy thoughts when the theologian wins. The winner of the debates should be the best debater regarding certain skills (wit, knowledge, debating, speaking, public speaking [belief and rationalism?] - just some skills which could be good for a debater).
This will require more nobles. Mostly, except the head of the university, they shouldn't require extraordinary quarters.
Alternatively, if we could put children in cages or force nobles into labour camps i would be equally happy.