The only son of the wealthy and respectable Bungo Baggins and Belladonna Took, Bilbo grew up at Bag End, which his father had excavated in Hobbiton Hill. He seems to have visited his mother's family at Great Smials, for he remembered Gandalf's firework displays there.
Bilbo's father Bungo died in 2926, and his mother Belladonna in 2934, after which Bilbo lived alone at Bag End for seven years until a fateful morning in the spring of 2941, when his pleasant, uneventful life was disturbed by the arrival of Gandalf.
More than a hundred and fifty years before Gandalf disturbed Bilbo in his garden that spring morning, the dragonSmaug had descended in flame on Erebor, the Lonely Mountain, an ancient kingdom of the Dwarves far to the east of the Shire. Those Dwarves that survived scattered throughout the northern lands, and many, led by Thorin, the rightful heir to the kingdom, travelled into the west and settled in the Blue Mountains near the borders of the Shire. Now, with Gandalf's aid, Thorin had begun to plot his revenge on Smaug.
Gandalf convinced Thorin that a Hobbit should accompany his party, both because of the Hobbits' natural stealth, and because he wanted to involve them in the wider world, as a preparation against the coming darkness. He chose Bilbo largely because of his Took inheritance, but he later suggested that Bilbo might in some sense have been 'fated' to go on the Quest.
So it was that on an evening at the end of April12941, Gandalf returned to Bilbo's house with Thorin and his twelve companions, and the Quest of Erebor was begun.
In Chapter 2 of The Hobbit (Roast Mutton), Bilbo and the Dwarves set out 'one fine morning just before May' (our italics), and in The Quest of Erebor (in Unfinished Tales) we're given an even more precise date: 27 April. However, when Bilbo looks back on their departure while travelling through Mirkwood (Chapter 8, Flies and Spiders), he remembers that they 'started their journey that May morning long ago.' Perhaps the rigours of his long journey had affected Bilbo's memory!