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I have just unlocked rituals, and I have absolutely no idea what I'm supposed to do with them. I spam my spells to no apparent effect, get shot at by a book, and lose 10k thaumaturgy. What am I supposed to do to maintain stability? What am I supposed to do to progress to the next level?

Learn the Fireball spell (you need to discover "Dragon") and get Blaze as a resource. Equip a few Fireballs into your loadout before you start combat, and one Conjure Blaze. (You will get more combat spells later on.)
Afaik the only way to recover stability mid combat is with potions.