Alphinaud suggests contacting Cid about finding a way into the Aery.
This quest is available if you have completed the quest The Wyrm's Lair.
Starting the Quest
Talk to Estinien in the Churning Mists to begin the quest.
Speak with Alphinaud
Go back to Ishgard and use the Aetheryte to go to the Skysteel Manufactory Aethernet Shard. Talk to Alphinaud to proceed.
Speak with Emmanellain
Use the Aethernet Shard to go to the Last Vigil Aethernet Shard. Talk to the House Fortemps Guard to enter Fortemps Manor. Talk to Emmanellain.
Speak with Cid
Go outside and use the Aethernet Shard to go to the Athenaeum Astrologium Aethernet Shard. Go south to the orange circle on the map. Cid is in the northwest part of the orange circle. Talk to Cid and there will be a cutscene.
Speak with Biggs
Go northwest to the Aethernet Shard and use it to go to the Skysteel Manufactory Aethernet Shard. Go northwest to the small flight of stairs, and go clockwise around the Skysteel Manufactory building. Talk to Biggs to complete the quest. There will be a cutscene. This one is important, so it is recommended to watch it either now or via the Unending Journey in an inn room.
Next Main Scenario Quest
After you complete the quest, talk to Tataru to accept the next quest: A General Summons.