The Fellowship of the Ring Book 1, Chapter 2
The disappearance of Bilbo is talked about for a very long time in the Shire. Every passing year, Frodo throws a party for Bilbo even though others assume that he is dead. Frodo lives by himself in Bag End and seems, to his countrymen, to stop aging.
"He found himself wondering at times, especially in the Autumn, about the wild lands, and strange visions of mountains that he had never seen came into his dreams. He began to say to himself 'Perhaps I shall cross the river myself one day.' To which the other half of his mind always replied 'Not yet.'" Book 1, Chapter 2, pg. 66
With age, he gets more restless. There are rumors of strange things about in the world. Many dwarves move over the earth. Some people speak of Mordor. An evil power had been driven from Mirkwood and founded a new home there. Orcs and Trolls are multiplying. Elves have been seen leaving through the Grey Havens in the west. It is early April and Gandalf appears for the first time in nine years. He looks older. He sits near Frodo and thinks of the 80 years before when he dropped in on Bilbo. Gandalf tells Frodo that his Ring has great powers; it was one of the great Rings of power. It makes its bearer invisible and gives him a longer life but eventually devours him. Bilbo's health and youth came from it and Frodo still looks so young for the same reason. Bilbo hadn't understood it but Gandalf had suspected the ring for a long time. He had tried to consult Saruman the White to no avail. Gandalf feels bad for Frodo because he was responsible for him being given the burden of the ring. He who keeps it desires it more with each passing day. The only reason someone has not come for the ring is that the powers of evil have overlooked hobbits and the Shire.
Sauron the Great, once ruler of evil and once deposed, is rebuilding his forces in Mordor. He has the nine Ringwraiths already enlisted working for him. In the beginning there were nine rings of power for men, three for elves and dwarves. Sauron learned how to craft the ring to rule it all. He lost it and now wants it back. It was cut from his hand by a leader of men who lost it in a river. Years later, a hobbit-like creature found the ring. His friend, Smeagol asked for it and when he didn't give it, Smeagol took it by force. This creature began to fish, hunt and steal, always wearing the ring. Eventually, he descended below mountains to avoid the harsh light of the sun. His body and mind were warped by extended use of the ring. This creature became the Gollum from whom Bilbo took the ring. Frodo balks at the idea of such a creature being related to hobbits. Gollum was never truly happy with the ring but became wretchedly unhappy without it. Gandalf thinks that the Ring left Gollum on purpose to be carried by Bilbo. He thinks it was trying to get closer to its maker. Frodo doesn't believe the tale, but Gandalf says he met Gollum in Mirkwood. The creature was searching for Bilbo so that he could get his ring back.
Gandalf's friend Aragorn found Gollum coming back from Mordor. He had told Sauron the name of the Hobbit who took the ring and where to find him. Frodo says it is a pity that Bilbo did not kill Gollum. Gandalf says "'Pity? It was pity that stayed his hand. Pity and Mercy: not to strike without need.'" Book 1, Chapter 2, pg. 85. The Elves of Mirkwood have Gollum in prison. The only place to destroy the ring is where it was forged in Mordor. He tells the Hobbit that for better or worse he was chosen to bear the ring. Gandalf will not take the ring himself because the temptation to wield its power would be too great. He asks Frodo to make a decision. Frodo thinks he should leave the Shire so that nothing bad happens to it. He wants to follow Bilbo and Gandalf is impressed with the lucidity of his decision. Frodo takes a new last name, Underhill. Gandalf tells him to choose a companion carefully and looks out the window to find Sam eavesdropping. Frodo tells him that he must keep it secret and Gandalf tells him that he is to go on the journey.