How can I upgrade schools?

Schools do not work that way.

As you play the game, you'll get Recipe drops fairly often. Recipe Drops are tied to a certain school - Things like Spears, Shields, and Light Armor are tied to the Cavalry school, Bows and Arrows are tied to the Archery School, and Swords/Axes/Medium Armor are tied to the Infantry school (Many more examples besides this).

When you learn those recipes, you can then make the item at said School.

Check out https://search.projectceleste.com/?tab=3&search= and type in the name of the school (Infantry School, Cavalry School, Construction School, ect) to see what all recipes are available for that school.

Keep in mind that you can only have 2 schools active in any Civ. Additionally, there is a school that can only be used by the Celts, the Metalworking school, which makes many consumables along with Celt-specialized gear such as Torcs, Warpaint, and Horns. Thus on your Celt civilization one of the two schools you use should always, 100%, be the Metalworking School.