Bilbo and Gandalf's first conversation is filled with a lot of funny snark from both parties.
Bilbo: Good morning!
Gandalf: What do you mean? Do you wish me a good morning, or mean that it is a good morning whether I want it or not; or that you feel good this morning; or that it is a morning to be good on?
Bilbo: All of them at once.
Bilbo: Good morning! We don't want any adventures here, thank you! You might try over The Hill or across The Water.
Gandalf: What a lot of things you do use "Good morning" for! Now you mean that you want to get rid of me, and that it won't be good till I move off.
In the beginning, Bilbo, expecting Gandalf, finds his home invaded by progression of dwarves. At the very end, he hears someone banging on his door and rushes to open it. The last four dwarves, including Thorin, tumble in and onto him, while Gandalf, safe in the back, laughs his head off.
The Company's meeting with Beorn. Gandalf craftily only introduces them two at a time while telling their story to keep Beorn's interest and make him more amenable to having all fifteen of them as guests. Particularly funny is Beorn's constant confusion whenever Gandalf slowly increases their number in the story to introduce each pair of dwarves:
Gandalf: I was coming over the mountains with a friend or two...
Beorn: Or two? I can only see one, and a little one at that.
Gandalf: ...when the goblins came down from the hills and discovered us. They yelled with delight and sang songs making fun of us. Fifteen birds in five fir-trees...
Beorn: Good heavens! Don't pretend that goblins can't count. They can. Twelve isn't fifteen and they know it.
On their second day at Beorn's place, both he and Gandalf disappear for the whole day. When Gandalf finally returns they naturally ask for an explanation, but he insists on having (a very large) dinner first. He finally starts to tell them...until he realizes that Beorn's hall is perfect for smoke rings and forces them to wait again while he blows multi-colored smoke-rings all around the hall.
Beorn cheerfully complimenting the dwarves on their story of murdering the Goblin King, and he liked it even better finding out they were telling the truth.
When Bilbo uses the Ring to rescue the dwarves from the horde of Giant Spiders, he sings a ridiculous song off the cuff to further confound them. Then the narration takes a moment to specifically point out that the word 'attercop' is particularly enraging to spider folk, but never explains why. * Given Tolkien's interest in linguistics, it may be because the word's literal meaning is "poisonhead", which does sound rather insulting.
In the Latin translation, "Attercop" is rendered as "Arachne".
When Thorin is captured in Mirkwood by the Elvenking and interrogated, he avoids all the questions about why he and his Company were in Mirkwood and trespassed into the elves' lands by repeating that they were starving and lost. Finally, when the Elvenking asks in exasperation what they were doing in the forest in the first place, Thorin sarcastically says they were looking for food...because they were starving. The Elvenking then orders Thorin to be thrown into the dungeons out of sheer frustration!
When Bilbo concocts the plan to leave the wood-elves' cave via barrels sent down the river, the dwarves are understandably leery and grumble about it. Bilbo then sarcastically suggests taking the dwarves' back to their safe cells and locking them back in, so they hastily agree.
And after Bilbo sends the dwarves into the barrels, he realizes that he forgot to think about escaping himself. The Lemony Narrator points out that savvy readers most likely saw that he Did Not Think This Through, but defends Bilbo by stating that they wouldn't have done any better in his shoes, as it were.
After Bilbo and the dwarves make it to Lake-Town, the dwarves are cheered up considerably about being treated as honored guests by the Lake-Town citizens, but poor Bilbo caught a cold due to having to ride on the barrels in the river. When the dwarves make speeches in his honor, all he can say in reply is "Thag you bery much!"
And it was around his birthday too, as the narration mentions and as Bilbo himself recalls later!
When the Company finally reaches the Lonely Mountain and finds the side door, Thorin begins to give a speech about how it's time for Bilbo to perform his duties and sneak in to Smaug's lair for them. Bilbo by then is so used to Thorin's speechifying that he cuts in.
The conversation with Smaug. All through the book Smaug has been talked about as this terrifying, roaring predator, a cunning and malevolent beast like any other fictional monster; but instead, when Bilbo goes down into his hall a second time, you discover Smaug's not a dumb brute at all - he's a cad! His self-satisfied eloquence, his wordplay and double-speak, the joy with which he picks apart Bilbo's riddles, are all hilarious.
Bilbo: I come from under the hill, and under the hills and over the hills my paths led. And through the air. I am he that walks unseen.
Smaug: So I can well believe, but that is hardly your usual name.
Bilbo: I am the clue-finder, the web-cutter, the stinging fly. I was chosen for the lucky number.
Smaug: Lovely titles! But lucky numbers don't always come off.
Bilbo: I am he that buries his friends alive and drowns them and draws them alive again from the water. I came from the end of a bag, but no bag went over me.
Smaug: These don't sound so credible.
Bilbo: I am the friend of bears and the guest of eagles. I am Ringwinner and Luckwearer; and I am Barrel-rider.
Smaug: That's better! But don't let your imagination run away with you!
The first time Bilbo went down he found the dragon at once but managed not to wake him. Then he figured he may as well start earning the title of his job, so he takes a single gold cup out of the massive pile of gold and jewels to take back as a trophy. Smaug notices the cup is missing, and flies into such a rage over it that he might have started bringing down the mountain!
This is actually continued with asides in The Lord of the Rings about the particular focus on manners and legal practices in the Shire.
At the end, when Bilbo is returning home with Gandalf and the elves of Mirkwood, he gives Thranduil a beautiful necklace when they part. When the Elvenking confusedly asks why Bilbo is giving him such a fine gift, considering all that Bilbo did before the Battle of Five Armies, Bilbo sheepishly replies that it's to pay for the food and drink that he stole while he was hiding in the king's palace in Mirkwood.
The Elvenking, in return, adds a wry aside to his otherwise ceremonial farewell to Bilbo:
"I will take your gift, O Bilbo the Magnificent," said the king gravely."And I name you elf-friend and blessed. May your shadow never grow less (or stealing would be too easy)!"
The reason Gandalf wanted to add Bilbo to the party is because Thorin doesn't like Hobbits. Apparently this bothered him so much (or the possibility of future conflicts amused him enough) that Gandalf just had to have Bilbo as their burglar.
The ultimate punchline; When Bilbo finally returns to the Shire, he doesn't get a hero's welcome. Instead, he's been declared legally dead (from the Shire's point of view, he did vanish without a word as to where he went about a year ago) and his estate is being auctioned off! He shows up just as his cousins are measuring his hobbit hole to see if their furniture will fit, and few people (particularly, the ones who got a good deal at the auction) were willing to believe that it was really Bilbo. He has to resort to buy most of the furniture back because simply buying it was easier than proving he’s actually alive in front of the law.
Gets even funnier when you read Lord of the Rings: Those cousins who were measuring out Bag End? Sackville-Bagginses.
During the otherwise nightmarish Riddles in the Dark scene, Bilbo asks a riddle that stumps Gollum. Gollum makes angry hissing noises as he tries to figure it out, and Bilbo gets this bit of snark in at him.
Bilbo: The answer's not a kettle boiling over, as you seem to think by the noise you are making.
The part where Bilbo climbs to the very top of a tree in Mirkwood to see where the sun is (so they can know which direction they've been travelling in) and sees a majestic view, complete with butterflies, is iconic and is memorably represented in both the animated and live action film. Neither adaptation has the following scene, where Bilbo climbs back down and gushes to the dwarves about what he saw. They're all irritable since they've been traveling through bug-infested dark woods for several days and don't share his enthusiasm (to say the least).