After you rescue Biggs and Wedge and the Tiny Bronco from Imperial scouts, you return to Maelstrom Command for your induction into the Grand Company.
This quest is available if you have completed the quest The Company You Keep (Maelstrom).
Starting the Quest
Talk to the Storm Personnel Officer in Maelstrom Command near the Aftcastle aetheryte in Limsa Lominsa to start the quest. There will be a cutscene. You can respond however you wish.
Speak with the Scion
Use a Vesper Bay Aetheryte Ticket (which you received after completing the quest The Scions of the Seventh Dawn) to return to Vesper Bay. Go into the Waking Sands and talk to the Scion of the Seventh dawn. There will be a cutscene.
Get Your Chocobo
If you haven't already, go to Limsa Lominsa and use an aetheryte to go to The Aftcastle, then go east to Maelstrom Command and talk to R'ashaht Rhiki to start the quest My Little Chocobo (Maelstrom). Now that you have completed "Till Sea Swallows All", you have enough Storm Seals to complete the chocobo quest. (See the quest page for the walkthrough.)
Next Main Scenario Quest
After you complete the quest, talk to Minfilia to accept the next quest: Sylph-management. There will be a cutscene.