I am about a third of the way through with the second book. And I have also gotten to the end of the HBO series of A Game of Thrones. I am pleased to report that the series on HBO was very similar to the book, and followed it quite nicely. The characters, specifically the Stark children, are presented as being slightly older in the TV series than they are in the book. However, the difference is minimal.
Some new characters have been introduced in the second book as the race for the rule of the seven kingdoms plays out. Stannis Baratheon, brother to the late King Robert Baratheon, believes he is the true heir to the throne and has been driven to extreme lengths to garner power. He has no denounced the old gods and has taken up his wife's religion, where Melisandre is the primary god. Melisandre has promised him strength and power against King Joffrey in return for his allegiance. A power-hungry Stannis agrees.
Also, Arya, under the guise of an orphan boy, is being led to Winterfell by Yoren. However, the Queens guard has intercepted them on their route and pronounced their need for another young boy whom Yoren has taken in his watch. Arya cannot comprehend why the Queen should be looking for a commoner boy when she herself is also missing. Martin leaves trace details about the boy, other than the fact that he was a steel-makers apprentice. From past knowledge, the reader can deduce that this boy was one of the many children of the late King Baratheon. And Queen Cersei will do anything to be rid of all of his illegitimate children, despite the minimal threat they pose to the throne.
Lastly, Tyrion has taken up the post as the King's hand. He is secretly trying to right the many follies that his nephew King Joffrey has commited. Tyrion realizes that Joffreys consultants and those that guide him in all of his major decision-making are leading him astray in order to settle their own scores. Tyrion is proving to be a Lannister that many can count on.