Galfrid needs to make sure you are properly geared for the tasks you may need to fulfill.
This quest is only available if you started the game in Gridania, you have completed the quest To the Bannock, and have reached level 5.
Starting the Quest
Talk to Galfrid in the Central Shroud to start the quest.
Upgrade your Gear
Galfrid won't let you proceed until you have equipped your head, body, hands, legs, and feet with gear of item level 5 or above. Most of your starting gear is already item level 5, but you may need to get a new hat and weapon. You can get a new hat and weapon for free by doing your level 5 job quest. Otherwise, there is a Merchant & Mender nearby. When browsing their goods, put the cursor on an item to see its details. The Item Level of each piece of gear is listed below the Defense and Magic Defense.
Present Yourself to Galfrid
After you have equipped the proper gear, Galfrid will have a green checkmark over his head. Talk to him to complete the quest.
Next Main Scenario Quest
After completing the quest, talk to Galfrid to accept the next quest: Chasing Shadows.