What are Bilbo's feelings as he and his friends set off for Lonely Mountain? How do the travelers get the door open?

They are treated like royalty in Lake-town, but they are also eager to leave for Lonely Mountain. The travelers get the door open after Bilbo sees a thrush and remembers the riddle on the map. He then knows that they have to wait for the sun to fall and illuminate the perfect spot on the rock in which to put the key. When they accomplish this, they have the outline to the door.