Galfrid needs you to investigate a situation at a dungeon called Spirithold.
This quest is only available if you started the game in Gridania, and you have completed the quest A Soldier's Breakfast.
Starting the Quest
To start the quest, talk to Galfrid in the Central Shroud. There will be a cutscene.
Speak with Alestan
Make your way southeast to the destination marked on the map and talk to Alestan.
Rescue the Missing Conjurers and Wood Wailers
Each of the five missing people will be marked on the map. Go through the gate into Spirithold and fight your way past monsters, and talk to each of the five people with the quest icon over their heads.
Rescue the Hearer
After you have helped all five people, go through the gate and you will see the Injured Hearer. There will be a glow around him, indicating another instanced battle. Check on him and there will be a cutscene, and then the battle will ensue.
Solo Duty: Spirithold Broken
In this battle, a mysterious black-robed figure will summon a Clay Golem. During the battle, the golem will use the attacks True Grit and Plaincracker. True Grit is a cone attack, and Plaincracker is a circle attack centered on the boss. To avoid these attacks, step out of the orange area before the enemy's cast bar is full.
After the battle, there will be a cutscene. This one is important, so you should probably not skip it.
Report to Alestan
Leave through the gate and make your way out of the Spirithold. Talk to Alestan just outside of the Spirithold.
Speak with Miounne
Return to Gridania and talk to Miounne. There will be a cutscene. Afterward, inn rooms and levequests will be unlocked.
Large cities have inns that you can enter for free by talking to an innkeeper. They are marked on the map with a bed symbol. In the inn room, you have access to a Summoning Bell, Crystal Bell, Toy Chest, Glamour Dresser, Armoire, The Unending Journey, a bed, and an Orchestrion.
Some of these features are unavailable for now, but you can use The Unending Journey to re-watch cutscenes, and you can use the bed to /doze on (i.e. use the Doze emote on) or use it to log out or exit the game. (You can also log out or exit the game anywhere via System in the Main Menu.)
Levequests provide a way to earn experience, gil, and miscellaneous rewards. Talk to Gontrant in The Carline Canopy to unlock them. From then on, you can talk to him to see the list of available levequests.
The Recommendations window is also unlocked. You can open it via the main menu, in Duty > Recommendations. You can select "All" to see all recommended activities in your current area, or select a specific item from the drop-down menu. For example, in an overworld area, you can select FATEs to see which FATEs are active in your area that are equal or lower than your current level. The Recommendations window can be helpful if you are not sure what to do next.
Next Main Scenario Quest
After completing the quest, you will need to be at least level 10 to proceed to the next quest. At level 10, if you have completed your level 10 job quest for your current job (available in your job's guild), you can unlock other combat classes by talking to the Guild Receptionists in the various guilds in Ul'dah, Limsa Lominsa, and Gridania. After you reach level 10, talk to Miounne to accept the next quest: On to Bentbranch.