You can view and level your Spirit Animals by pressing after beating a level or simply choosing the left-most option at the title screen. It costs one Spirit Badge to increase an Animal's level from 1 to 2, two badges for the next level, and so on. 10 is the maximum level, so you'll need a total of 45 Badges to max an Animal out.
Standard Adventure levels offer three badges and Adventure bosses offer one. If you manage to earn every badge for every Adventure level you play, then you'll be able to max out your first Animal, the Spirit Monkey after beating 2-5. Don't worry if some levels' Badge requirements give you trouble at first. Come back later after you've built your Animal(s) up a little more and you'll have an easier time.