So this is medieval high fantasy world where kids go to school, which does not cost money. Magic is practiced regularly. Realistically, would kids go to magic school? If not, where would they learn it? It could be possible to learn it in university later on, or join a guild of magic, or not at all. Thanks
As the author, it's completely up to you, especially since "high fantasy" can stray rather far from reality.
What I can say is that formal public education was very rare in actual medieval times, and when extant, mostly intertwined heavily with religion. Low and middle class individuals almost exclusively learned in master-apprentice settings with only the very wealthy and nobles attending actual learning institutions (typically churches) or having private tutors.
Unless your society is rather modern, I'd expect several magic institutes or universities to exist that are attended by only the progeny of the ultra-rich and powerful. Everyone else would learn magic from a master (the village witch or from a guild member or whoever).
Teaching and controlling magic? Hell Yes.
Schools? Depends on the setting.
Schools require a more structured academic approach that won't fit in much with Medieval Europe. Now that is not the case everywhere else in the world.
Anyway schools might become a necessity for wizards because obviously you want to teach and control the people with power to cause a lot of problems.
You can't form a militia without your lords consent so you sure as hell can't run around with the power to murder and rape at you fingertips without your lord having a say.
This can lead to many scenarios which I will touch lightly on because not a lot of context is provide and this question will probably be close.
Dragon Age I and II style. The wizards are feared, loathed, mistreated, and border line tortured and constantly abused by the templars and the chantry. This is somewhat explained actually as the number of blood mages and wizard turning into demons in that world is absurdly high. It's like a blood mage using it and turning into a demon is a stable in the world or something.
But that won't remain for long as people with a lot of power would eventually rebel against constant and unjust oppression.
Thought it is your setting. Maybe your own thing is more NobleBright than this.
On the other hand this could lead the wizard themselves into organizing themselves and teaching magic.
Again if you are taught how to wield a spear then wielding power over life and death is more dangerous and requires more supervision and all that.
So a bunch of wise old wizards, big beards with cloaks and a grumpy attitude is a stable, decided to make schools and teach wizards.
Magic schools are inherently problematic but that is your role here. To figure out a realistic half working or fulling working system.
Because give teenagers normal libido and magical powers and you have a recipes for disaster.
But again schools are better as they at least control and guide and teach children.
Effective measures of control and tricks of how to handle them is completely different.
Interestingly this could lead to a rise in teaching institutes as people who do other things, medicine or metalworking or whatever, would copy this approach of wizards. So you could actually accelerate the history of the world and the science develop faster.
Another approach is less structured less modern approach.
It is the duty of the local court wizard to gather wizards and teach them in an apprenticeship kind of way.
This insures that the new wizards are loyal to the local lord and also since it is the duty of a person in court then they have the support of the monarchy/duke/lord to seek out and teach those children how to use magic.
So lets assume lord Vargin has 30 cities. His court then have a dozen wizards most of them are given the repeatability of collecting and teaching young wizards to do what needs to be done, whatever that is.
So once they hit the town they already in their own lands and get the local gossip. The blacksmiths daughter is seen doing magic but he is denying it. In that cabin on the edge of the forest that guy is totally raising the dead to play football matches. They are quite good. But guess I prefer this bag of coin. So local knowledge and the resources of the state would help.
In both cases, schools and apprenticeships, there needs to be certain things
First wizards and their families should be inclined to join the system. Otherwise it is easy, unless your setting has it differently, for parents to just hide their children.
So money or good jobs or whatever should be expected as a reward for the parents.
Children should be excited and happy to join a place that is built it to be the best thing ever for wizards.
It's like NASA if you love space.
So the prevailing view that it is an honor to be taught magic and they willingly go and do it.
Then you should not abuse them much. Sure you can do it. But depending on the setting that might lead to one angry wizard that got whipped one too many times to overthrow the entire kingdom and become an immortal warlock with a penchant for regicide.
I mean that is something I really don't get. Why anger people who can incinerate you in a heartbeat?
Anyway please provide more context so we can answer it in a better way