When Bilbo found the ring of power on the ground in Gollum's cave it seemed to have no effect on his emotion. But as he learned of its power a small change seemed to happen. This was evident when at the entrance to the goblin exit he gathered all his muster and took a leap over Gollum's head. The old Bilbo would probably not have tried that and would have eventually been caught sniveling and frightened in the corner somewhere. After he escaped the cave he eventually met up with his party and there his bravery led them to trust Bilbo.
After the escape from the goblin tunnels when the dwarves heard the story of the riddle game and Bilbo squezing through the door they had new found respect for the hobbit. I believe this confidence helped Bilbo emerge from his calm and rational self and led him to speak his voice and take more charge in decision and action times. This is very evident in the battle with the spiders as he acts very diffewrently than the origanal Bilbo by swinging from trees, cutting loose his friends, stabbing spiders and talking trash to his enimies. This could not have been more different, and I believe it was becuse he knew that he party believed he could it.