After you attend the remembrance services in each city-state, you choose to join the Twin Adder.
This quest is available if you have completed the quest A Hero in the Making.
Starting the Quest
Talk to the Serpent Officer in the Waking Sands in Vesper Bay in Western Thanalan and say Yes, then accept the quest.
Speak with the Personnel Officer
Go to Gridania and and use an aetheryte to go to the Aetheryte Plaza (if you are not there already). Go west to the Adders' Nest. Talk to the Serpent Personnel Officer. There will be a cutscene. You can respond however you wish to the question.
Determine the Fate of the Airship and its Crew
Use an aetheryte to go to the Lancer's Guild, then go east to the pier to the Westshore Pier and talk to Romarique to purchase passage to Sweetbloom Pier (it's free). Then go southeast (attuning to the Hawthorne Hut aetheryte if you haven't already). The destination is southeast of the Hawthorne Hut.
Speak to the Airship Crewman. There will be a glow around him, indicating an instanced battle. Check on him and there will be a cutscene, and then the battle will ensue.
Solo Duty: The Company You Keep (Twin Adder)
Attack the Imperial Funditors and Imperial Imaginifiers first, then the two Bestiarius, and finally the Decurion. When the Magitek Vanguard appears, go off to the side, because it will sometimes target a random ally with Cermet Drill, a line attack, and if you stand off to the side, you can bait the attack away from your allies. Defeat the two Funditors that appear at the same time as the Vanguard, then finally attack the Magitek Vanguard until the battle ends.
Following the battle, there will be a cutscene.
Speak with the Personnel Officer Again
Go back to Gridania and and use an aetheryte to go to the Aetheryte Plaza (if you are not there already). Go west to the Adders' Nest. Talk to the Serpent Personnel Officer to complete the quest.
Get a Chocobo
It is strongly recommended to talk to Vorsaile Heuloix to start the My Little Chocobo (Twin Adder) quest. Upon completing this quest, you will receive your own chocobo mount. However, you should finish the next quest, Wood's Will Be Done, before you proceed with My Little Chocobo, because the quest reward will help you complete it.
Next Main Scenario Quest
After you complete the quest, talk to the Serpent Personnel Officer to accept the next quest: Wood's Will Be Done. There will be a cutscene.