Wall

Tim, frustrated with his life as a penniless and impoverished outcast, decides to leave King's Landing, always wanting to go to the Wall and face the horrors and stories and legends. Thus, he heads north along the highway and follows it to the Wall.

He does not stop at any village or town, although he does rest again in the city of Winterfell after approaching the Wall. After that, he heads straight for the Wall.

When he reaches Castle Black, he convinces the guards to let him through the gate. It takes some time, but he manages to get them to let him through, although they warn Tim that depending on the conditions of his return, they may not be able to let him back in. But it's a risk Tim is willing to take.

Now that he's north of the Wall, the real problems begin. He has to decide where to go, and soon. He can always go straight to Craster's Keep in the Haunted Forest. Or you can just keep going.