What do the bond level and stages mean?

Each animal you've captured has one bond with you, shared by every card of it. It grows as you talk, learn its lesson, put it into practice and change something in your life with it. The stages run Met, Getting to know you, Working together, Long-standing, Trusted and Mastered. Lists and chips show the stage. The card's Bond tab shows the bond level and XP, and the three strands the bond grows on: knowing the lesson, putting it into practice, and changing something in your life.