Perhaps teaching and so on is handled by the respective guild associated with the profession e.g. mason's guild members teach children. It could be possible to send certain children to "join the Mason's guild" and so on so they are taught in the ways of stoneworking and become adequate, novice or competent, skilled or proficient masons by the time they leave school when they grow up, depending on how well (and for how long) they were taught. Obviously, if you send an 8 year old off to the Blacksmith's guild to become an apprentice, he will be more highly skilled by the time he leaves school compared with someone who entered the guild at 10, simply because the 8 year old had more training. This would hinge on the personality of the dwarf children of course - they may not like learning, they may be a bit slow or they may not have any natural inclination to the job (try sending a musical, creative child to become a miner), so they may leave school as "adequate" or even "novice" masons when others leave at "competent". That would encourage the player to examine dwarf children more carefully to see where their talents may lie. For the lazy players, maybe the manager could do that for you so it's automated.
Tsen also raises a good point - perhaps the guilds would teach each dwarf child to read and write as part of their training. The dwarf children would gain a small note in their P/Z description saying that they are an apprentice mason or an apprentice miner etc, and also a member of the "miner's guild" or the "mason's guild". Another possibility is that the religions and temples may teach dwarf children to read and write, so their education runs in tandem with the guild and the "church".
I like the idea of sending dwarf children off to the army to become squires too. Similar rules could apply to them, including preferences. Perhaps squires could be assigned to certain warriors so that they could learn certain skills from them. As the religious organisations would teach the children to read and write, the squires will not lose out. This goes a bit deeper than simple training because your legendary spearman may have dabbled in wrestling and more unusual skills like biting, so the squire would gain most (if not all) of these skills, again depending on the teaching skill of the soldier. Imagine the spearman teaching the squire all the dirty tricks he knows, or unusual hand-to-hand combat tactics (as will be implemented soon). It's great how this idea is so easy to build on; you could create an adventurer and develop new wrestling moves, then join the fortress army, get a squire and teach him everything you know.
Another variable you could add to this is whether or not the child likes the teacher. If the child holds a grudge against the teacher, or vice versa, then the child will not learn as much. If the child and the teacher become friends, then the child learns even more, or may at least be less likely to reject their appointment as an apprentice of a profession they dislike. For example, there may be a very musical and artistic child who is assigned to the miner's guild. He rejects this assignment, so he does not learn as much and has unhappy thoughts. However, the friendly teacher means he is more comfortable with learning to be a miner, so his unhappy thoughts are ultimately negated, and he learns more.
Christ, this could get complicated.