Without the Cloud Stone, a fall from the Heavenly City would kill you. But with it, you can survive. And if you figure out the secret of the Castle in the Sky, you can use the Cloud Stone to gain access to an otherwise inaccessible castle in Fairy Land: the Water Palace.
Despite the difficulty to reach the palace, there isn't a ton to do when you first arrive. Nearly the entire right side of the palace is dedicated to the king's treasure collection, and a guard won't let you past to examine any of it. A woman in another room will mention that the collection contains the Scroll of Jeem, which is needed to reveal something.
Keep talking to other members of the Palace. There's a lot of talk about the cemetery, particularly about the need to read runes, and the fact that people are living under it. Another woman will exclaim that she's lost her identification card.
If you continue investigating the Palace, you'll discover the room known as the Sleeping Chamber off to the far left side of the Palace. The women there won't appreciate you walking around there, but keep your ears open for a sound that indicates that you've discovered something. When you hear it, squat down to examine the floor, and you'll find a missing ID card. If you attempt to return the card to the woman who lost it, she will claim she created a new card, which means the card you found is yours to keep. This card will now make it possible for you to come and go from the Palace, without the need to fall down from the Castle in the Sky every time.
The only other person to speak with is the king of the palace. The conversation will be brief. He'll ask if you have anything interesting, but conclude that you do not. Then he'll tell you that he's always wanted to see a dragon's fang. You probably haven't heard anyone mentioning a dragon until this point. With no other leads, your best bet is to explore the cemetery that people here are talking about.