Continually talk to city guards. Speak with inn or tavern keepers and ask if they have heard any gossip. Talk to the orphans at Honorhall Orphanage in the city of Riften.
Travelling on foot is the most adventurous and you can find alchemy ingredients along the way. Outside of most major cities, you can buy a horse for a faster trip. Book passage in the back of a carriage, usually found in the same area where you would buy a horse. You can also fast-travel to Windhelm if you have been there before.
As long as you do not attack anyone else in the orphanage, killing Grelod is not considered a crime.
If you don’t get transported to the abandoned shack, wait a few in-game days, and then sleep again.
You can speak with the captives and listen to their stories. Kill whomever you think deserves to die. Your decision has no impact on gameplay.
You can now take on killing contracts for money (usually a few hundred gold).
