Beat the game once. When playing NG+ The badge and costumes will be in a new "Items" tab in options.

Unlockable	How to unlock
Badge of Honor (Trico)	Beat the game once.
Enduring Attire	12 Barrels.
Fine Featherpainter	96 Barrels.
Homecoming Costume	24 Barrels.
Horned Apparel (ICO outfit)	48 Barrels.
Warrior's Clothes (Shadow of the Colossus outfit)	64 Barrels.