Thank you for playing!
You can't catch demons at all actually. You pick one (for now only one, but three are planned for the final game, they are basically done but have no graphics yet) and it gets upgraded from time to time via the story.
If you picked 'yes' on the tutorial part there _should_ be a tutorial talking about trophies as soon as you leave the first building after defeating the first boss, but I agree it could be explained or emphasized better. That's something we'll have to look into!
Thank you for the feedback! It's valuable.