There are several ways to level up your characters.
You can level up your normal attributes (health points, defensive points, etc.) by earning experience points or by eating Perfect Bug Milkshakes.
Your can level up your Rank by eating Monster Heart Cupcakes. When your Rank goes up, your special abilities are upgraded to the next level. Other benefits may occur, as well, depending on your character.
Going from Bronze to Silver to Gold: You can use a Special Receipe Pickle to change your character's status. You can get a Special Receipe Pickle by completing all of the Quests from a Quest Giver or you can buy them from the Shop. When you upgrade the status of your character new special abilities become available.
Special Ablities by Status:
Bronze: 1 special ability
Silver: 2 special abilities
Gold: 3 special abilites