The greatest development in Bilbo in the first eight chapters of The Hobbit is the emergence of Bilbo's self-confidence in his own abilities. He begins to see himself, not just as an ordinary hobbit, but as an adventurer--a burglar with skill and cunning. His quick and clever mind helps him out of scrapes and dangerous moments more than once. Bilbo learns to rely on himself and his own strength, rather than depending on the aid of others.
After Bilbo kills the great spider in Chapter 8 "all alone by himself in the dark without the help of the wizard or the dwarves or of anyone else," the hobbit realizes that he feels like a "different person, and much fiercer and bolder" (142). Bilbo's success in defeating the spider on his own helps him realize that he really does belong in Thorin's company as a worthy, contributing member.
The Hobbit by J. R. R. Tolkien is as much a coming-of-age story of Bilbo Baggins as it is a tale of adventure. In the first chapter of the story we see Bilbo as a contented domestic figure, interested primarily in food and comfort. The wizard Gandalf and the discerning reader, though, spot something amiss in this picture of the home-loving hobbit. There is a romantic streak to Bilbo and a curiosity about the world outside the Shire as well as a love for tales of adventure. Although Bilbo appears conventional on the surface, one gets a sense that Bilbo is complicit with Gandalf's manipulations of him even if he approaches his new life as thief and adventurer with some trepidation.
At first, Bilbo remains his luxury-loving self, enjoying the adventure as spectacle, concerned about inadequacy of food, and somewhat fearful and unsure of his own abilities. As the adventure progresses, he gains physical strength and endurance and increasing confidence in his own skills and cleverness. He also begins to assert himself more. The key episode in his character development is Chapter 5, in which he ends up left on his own and encounters Gollum. He ends up using his own wits to escape a terrifying situation. Even more importantly, at the end of the chapter, he is called upon to decide what sort of adventurer he will become with his new found strength, and chooses a path of mercy and kindness, letting Gollum live despite knowing full well that Gollum had planned to eat him.
The transformation of Bilbo from home-loving hobbit to an equal of the dwarves in their life of adventure is seen in Chapter 8 when Bilbo saves the dwarves from the spiders and realizes the degree to which he has changed and become braver and stronger.