One scene where Bilbo gains self-confidence is in chapter 9, when he helps Thorin and the other dwarves escape from the Elvenking's palace. Bilbo avoids being locked up by using his magic ring. The ring gives him the power to be invisible; while invisible, he finds where the Elvenking locked each of his friends:
Eventually, after a week or two of this sneaking sort of life, by watching and following the guards and taking what chances he could, he managed to find out where each dwarf was kept (chapter 9).
Later, Bilbo looks for a plan to help his friends escape. Bilbo carefully waits for the right moment for their escape. He eventually finds a way to steal keys from some elves who drank too much wine. Later, he helps his friends gain their freedom by hiding them in barrels to be floated down the river.
Another scene where Bilbo grows in his self-confidence is in chapter 16, when he decides to give the Arkenstone to the elves (for them to use as a bartering tool with Thorin). Bilbo explains to the elves that Thorin's love for his gold is great. He explains that he understands why the elves think they deserve reparations from the dwarves. However, he doesn't think Thorin will give in to their requests without added incentive. Therefore, Bilbo takes the Arkenstone, Thorin's most sought-after jewel, as his share of the treasure. He offers this to the elves, hoping that Thorin will be more willing to come to an agreement if the elves have the Arkenstone. Bilbo desires peace and is willing to sacrifice both his finances and possibly his friendship with Thorin and the dwarves to avoid war. Bilbo explains to the elves,
I am merely trying to avoid trouble for all concerned (chapter 16).
Bilbo grows in self-confidence and behaves increasingly bravely as the novel progresses.