Costumes on the main Islands are unlocked after the first Quest on the Colossingum has been completed. Some can be bought right away for Coins or Diamonds, while others are available at different times. When you perform actions on the Colossingum like completing a Quest, the cost of its corresponding Costume is reduced on those Islands.
There's a new way to collect Costumes: breeding! Whenever you're breeding for a Monster that can own Costumes, there's a small chance that the resulting Monster egg already owns a Costume! NOTE: Limited Costumes can also be 'bred', but only for as long as the Costumes are available to buy!