What are some quotes showing Bilbo Baggins's character development?

In a sense, Gandalf perceives depth to Bilbo's character that other people do not see. It is as much the case that Bilbo's potential has been limited by circumstances and that his adventure with the dwarves gives him the opportunity to fully live up to and into the potential as that he somehow changes character. 

At the beginning of The Hobbit, Bilbo appears to be a somewhat conventional hobbit who has assimilated the values...

In a sense, Gandalf perceives depth to Bilbo's character that other people do not see. It is as much the case that Bilbo's potential has been limited by circumstances and that his adventure with the dwarves gives him the opportunity to fully live up to and into the potential as that he somehow changes character. 

At the beginning of The Hobbit, Bilbo appears to be a somewhat conventional hobbit who has assimilated the values of his community, albeit with some minor eccentricities which eventually are revealed as part of his actually heroic nature. He expresses this superficial conventionality in the following statement:

“We are plain quiet folk, and I have no use for adventures. Nasty, disturbing, and uncomfortable things.”

As Bilbo listens to the dwarves, something of his hidden nature begins to awaken:

As they sang the hobbit felt in love of beautiful things made by hands and by cunning and by magic moving through him, a fierce and a jealous love, the desire of the hearts of dwarves.

As Bilbo embarks on his adventure, despite complaints about food and weather, he gradually reveals a sort of quiet determination and bravery. One major turning point occurs when he kills a giant spider in Chapter Eight:

Somehow the killing of this giant spider, all alone by himself in the dark . . . made a great difference . . . He felt a different person, and much fiercer and bolder in spite of an empty stomach, as he wiped his sword on the grass.

In Chapter Sixteen, Bilbo has done the moral and courageous act of bringing the Arkenstone to Bard. His transformation into a true hero is highlighted by the following statement of the Elvenking:

"Bilbo Baggins!" he said. "You are more worthy to wear the armour of elf-princes than many that have looked more comely in it. But I wonder if Thorin Oakenshield will see it so. . . . I advise you to remain with us, and here you shall be honoured and thrice welcome."