This hidden quest is too tough...!

I've figured out how to increase the 'Glory of the Academy,' but I have no idea how to raise my combat power, no matter where I tap! 💦 What is this 'Adventure' thing? How do I get there?? I'm only on the second stage, but it's so hard I don't think I can even reach the third. This is my first time in a dungeon, and it looks like clearing it will be tough too.