After you take care of an errand for Warin, Zuzumeda at Black Brush Station has a rumor for you.
Requirements
This quest is only available if you started the game in Ul'dah, and you have completed the quest Out of House and Home.
Starting the Quest
Talk to Zuzumeda in Black Brush Station in Central Thanalan to start the quest.
Speak with Wystan
Go south to the Coffer & Coffin, and go inside and talk to Wystan.
Give Cookies to the Children
Go outside and talk to each of the children with the quest icons over their heads, and hand over a Crumbling Ginger Cookie from the Key Items tab of your inventory to each child.
Speak with Wystan
After all four children have departed, go back inside and talk to Wystan. There will be a cutscene.
Follow Wystan to the Sil'dih Ruins
Go outside and make your way southwest to the quest icon on the map and minimap. There is a glow around Wystan, , indicating another instanced battle. Check on him and there will be a cutscene, and then the battle will ensue.
Solo Duty: Way Down in the Hole
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.
Speak With Wystan
Go back northeast to the Coffer & Coffin, and talk to Wystan.
Speak with Momodi
Go back to Ul'dah, either by going south from the Coffer & Coffin, or by using Return.
Once there, go to the Adventurers' Guild and talk to Momodi. There will be a cutscene. Afterward, inn rooms and levequests will be unlocked.
Inn Rooms
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
Levequests provide a way to earn experience, gil, and miscellaneous rewards. Talk to Eustace in The Quicksand to unlock them. From then on, you can talk to him to see the list of available levequests.
Recommendations
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 Momodi to accept the next quest: Takin' What They're Givin'.