Brienne is looking for Sansa. She is heading from King's Landing to Duskendale and makes it as far as Rosby in this chapter. Consulting my iPad map app, its not totally the wrong direction but it's not that right either. And she still thinks Dontos is with Sansa (he's at the bottom of Blackwater Bay somewhere). So unless she gets some better intelligence on Sansa's location it's going to be a long quest.
She goes around asking locals if they've seen people meeting Sansa's and Dontos' descriptions. Basically she's behaving like a player in a fantasy RPG computer game. If she can just talk to the right NPC she can progress the story. Instead she meets Ser Creighton the boastful and Ser Illifer the penniless. This seems like it is going to be a minor side-quest, though. She does learn that plenty of other people are also looking for Sansa, since Cersei is also trying to find her.
By the way what is a hedge knight? Some sort of lesser-knight, it would seem.
Edit: I found out. Hedge knight is a derogatory term for knight errant, which is a knight with no master who must seek work.
Interesting that the word on the street is that Tyrion is the cruelest Lannister. That's pretty far from the truth.
There is a group belonging to a strange religious sect heading for King's Landing. I wouldn't be surprised if this turns out to have more significance than appears here.
Brienne and the hedge knights ride to an Inn and Brienne pays for rooms for them all, but after going to bed and not sleeping she leaves them and gets back on the road.
