In the opening chapter of The Hobbit, Bilbo Baggins is depicted as being in a kind of conflict with himself. This theme is established early on, when Tolkien provides a digression concerning the Tooks, noting that Bilbo had inherited some of this unconventionality from his mother. There are several moments in this chapter where Baggins's own inward adventurousness is on display.

When Gandalf introduces himself, pay careful attention to Bilbo's reaction. He seems enthusiastic when he speaks of the various adventures Gandalf has been on. This isn't quite the same as wanting to go on an adventure himself, but it's noteworthy all the same. Later, note Bilbo's reaction to hearing the dwarven song ("Far over the misty mountains cold . . . "). We see that in this moment, Bilbo wants to join in that adventure. Tolkien writes:

Then something Tookish woke up inside him, and he wished to go and see the great mountains, and hear the pine-trees and the waterfalls, and explore the caves, and wear a sword instead of a walking-stick. (Tolkien, The Hobbit, chapter 1)

The above mentioned passage serves as only part of a longer descriptive passage. Clearly, we see that this song has affected him deeply.

Bilbo rejects the adventure at first, but changes his mind when the dwarves leave without him. 

We are told that the Baggins family is well-respected because they are responsible and never go on adventures.  Going on adventures is frowned upon in the Shire.  Hobbits don’t like adventures.  They like things to be dependable and quiet.  However, Bilbo’s story is a little bit different. 

This is a story of how a Baggins had an adventure, found himself doing and saying things altogether unexpected. He may have lost the neighbors’ respect, but he gained—well you will see whether he gained anything in the end. (Ch. 1) 

The reason Bilbo went on an adventure is that he is part Took.  Apparently the Tooks are a less respectable family, since they are open to adventuring.  The Took side of Bilbo took him on an adventure.  It helps to have wizards for friends.  They always get you into trouble. 

When Gandalf the wizard first told Bilbo that he was looking for someone to go on an adventure with him, Bilbo refused.  Then thirteen dwarves showed up at Bilbo’s door, had a great party, and talked a lot about a dragon and the treasure he was guarding.  They also commented that Bilbo looked “more like a grocer than a burglar,” and that upset Bilbo’s pride.  

So after that the party went along very merrily, and they told stories or sang songs as they rode forward all day, except of course when they stopped for meals. These didn't come quite as often as Bilbo would have liked them, but still he began to feel that adventures were not so bad after all. (Ch. 2) 

The dwarves actually turned out to be pretty considerate guests.  They cleaned up after themselves, and left in the morning.  Unfortunately, they left on the adventure without Bilbo. Bilbo had decided he wanted to go, without even realizing it. 

"Don't be a fool, Bilbo Baggins!" he said to himself, "thinking of dragons and all that outlandish nonsense at your age!" (Ch. 2) 

Gandalf told him they have left him a note on the mantle with terms, and so he ended up rushing out the door before he seemed to know he was doing it.  He left without his handkerchief! I imagine that is a big deal for a hobbit used to comfort, but it would soon become the least of his worries.