I may have a magic system worked out. I've tried to incorporate some ideas and suggestions, though if there's any questions or spotted issues, feel free to say.
On availability of magic. Essentially everyone receives a basic education about it and can do at least a small spell, even if it's just lighting a candle. For most this is the extent of what they can do. Maybe with hard work they can accomplish something like cooking food with their bare hands. Mages are defined by their ability to go further.
They can unleash gouts of fire and ice, enchant iron swords strong enough to break steel blades, summon creatures not of this world, and more. Mages are those with inherent talent who work hard enough and educate themselves enough to truly understand what magic is and how it works. There are levels to being a mage as well but even those with basic competency are valuable. Great mages are truly rare but are capable of great feats.
I know there was some worry about characters standing or being special. The trick to it in this case is that at our characters ages, they should not be competent in any form of magic. Even if they were found fitting to be mages, they should only just now be figuring out what they do well and what school of magic they wish to start specializing with(note: specialize in this case does not mean it is all they know. Basic mage schooling covers all types to a degree, but people have different things that they do best and can most easily learn).
Schools of Magic
Elemental: Probably the most visually dramatic yet simple, but also the most draining to use. At its heart, elemental magic is simply being able to cast the elements from ones self as a weapon. Controlling fire, ice, lightning, water, air, etc. and using it against an enemy as one sees fit. Mastery of the school brings higher intensity, scope, and control(able to engulf entire areas in hurricanes of sub-zero ice while the mage stands just outside and such). One of the more freely taught and accepted magics, considered essential for battles.
Restoration: A very narrow but extremely important school. It essentially encompasses healing and repair. Whether of people, clothes, pets, tools, anything that can be fixed. There isn't much variety among this school but it is considered one of the most important ones simply for the number of lives it can save as a good healer can treat wounds that would make most doctors prepare last rites. The best can resurrect the recently dead provided Eiryia has not yet claimed their soul. Healers are highly valued both in battle and at home, often having the full trust and goodwill of their community. Readily taught and encouraged in schools.
Runic/Enchantment: Possibly the most academic magic school in that it requires vast knowledge of the seemingly endless runes and enchantments, their purposes, and how they might interact. Reaching understanding though grants endless possibilities to what one can do. Enhance items, unlock things meant to stay closed, bind the living, dead, inanimate, even abstract ideas if one only knows the correct symbols/sequences and how to apply them. Typically done by carving into the associated target, but the best enchanters in history have circumvented this at times and even been able to draw into the air itself. Not quite encouraged because the sheer knowledge, patience, and attention to detail required of it is too much for most but experts are highly respected.
Druidism: A very old school of magic(often claimed to be the first taught to humanity by the pantheon) that focuses on manipulating the environment and dealing specifically with the scions and spirits of nature. The highest scale of this magic has been seen as controlling the weather or an entire land. Can seem rather old-fashioned and ritualized because of their customs, but often instills a great discipline in devoted practitioners.
Blood Magic: Not considered evil in and of itself, but powerful blood mages are watched with a wary eye and given the least amount of leeway before opinion sours on them. Barely covered in schools unless specifically requested and not many places have suitable teachers. Every spell is powered by blood, life, or both and the effects are often taboo. Necromancy, mind control, dream-walking, transmogrification of the flesh, all are the domain of blood magic. There is a certain irony though in that while none will say it, if a hard task needs done and a blood mage is on it, those involved will breathe a little easier. These mages will often cross lines and think in ways others simply won't, a polarizing but necessary trait in dire times.
Conjuration: At its most basic, a summoning school of magic. The variety of what is astounding though. From unique small animals to sentient engines of destruction, the possibility of what can be pulled from another plane of existence is limitless. Devoted followers become better both by practice as well as bonding with their summons if possible and benefit the most from having a Familiar companion. Readily taught and encouraged in schools.
I think that should cover most bases for schools but if I missed a big piece or if there are questions on what spells might fit in what category I will be happy to address as best I can. On the idea of having different experiences from different locations, that is possible. Most towns will have everything they can covered but do have to work with what is available. So if a place is a haven for druids, that will probably be a mage's specialty. If a person grows up with a blood mage parent who really wants to encourage their child in that direction then they'll probably be a blood mage. This is a list of what is done by regular governments provided they have the needed resources and expertise. If a place is lacking one way or heavily stacked in another, that can change the mage's education.
