Kaede stares some more at their Magical! Girl!! Magic!!! and it's so exciting and they can generate their own mana that is amazing. That is so incredibly amazing.
They can show Edie and Emily what their magic can do, now that mana isn't a scarce resource anymore. Basically there are three types of "active" mages, elementalists, arcanists, and enchanters, and the metamancers as a separate class.
An elementalist can have a single active magical effect on themself—flight, telekinesis, insubstantiality—which is called a blessing, and using a blessing depletes an individual per-blessing mana charge, which can only be recharged by having that blessing active but unused (or with metamancer help). Switching active blessings is an instantaneous primitive action that can be performed at any time the active blessing is not being used, and it's very intuitive. Blessings have various built-in safeties, and a mage knows how much mana they have left and what that will accomplish them.
An arcanist can bind magical effects to symbols, and a given magical effect is called a spell while the symbols used to activate it are its incantation. Spells are cast by spending mana, and arcanists are born with zero mana and start charging it up at a constant rate over time with no upper bound to the amount of mana that can be stored. For an arcanist to bind a spell, they need to define an effect they're going for and then attach it to symbols—actions or words or gestures—in a single sitting and thenceforth whenever the arcanist has enough mana and performs these actions the effect happens. It is also possible to write a one-use magic scroll for others to cast a spell. Spells are bound forever, except that a metamancer can unbind a spell.
Enchanters are like arcanists, except they can attach magical effects to objects, creating magical artefacts with various magical effects. These have a mana charge that runs out with time and can be recharged by any enchanter.
Metamancers do not generate their own mana, but they can store any of the other three types, and if they have enough can create any of the other effects. They can also perceive magic and drain mana from artefacts, scrolls, or other mages.
But Galatea magic Kaede already knows; what they can't wait for is the magic lesson. They have that nagging suspicion that all magic here's the same, but they can't really say whether it's true or not before they actually see all three kinds of local magic and compare.