The Dothraki are very important in the Game of Thrones storyline. They are obviously feared by those in Westeros, which we see when Robert Baratheon is adamant that Daenerys must die. They are also Daenerys's second home. They are also her protectors. She has been welcomed into the community as the Khaleesi and is respected as Khal Drogo's wife. This causes some anger and fury in her borther, who lashes out as an important ceremony, ultimately leading to his death. This shows how protective the Dothraki are of their leaders. It is also important for Daenerys to have the support of the Khalisaar to show those who wish her harm that she is well protected.
The East and West comes into play when Robert is talking about killing the girl. Ned states that she isn't a threat, but Robert is very concerned about a Targaryen having a "Dothraki Hoard" and potentially leading the Dothraki to the kingdom gates. This is also the binding of the two regions. Without Daenerys, there really isn't a tie between Westeros and Essos. She is the one originally from Westeros that is now serving as a leader in Essos, and posing a threat to Westeros in the King's eyes.
Overall, Daenerys's story is my favorite in the first season because there is such character building and we get to compare where she started out to the person she grows to be.