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Sméagol and Déagol go fishing in the Gladden Fields.
Déagol finds the Ring, which Isildur dropped about 2,500 years before when he was killed by Orcs.
Sméagol demands the Ring from Déagol.
Déagol refuses, so Sméagol strangles him and takes the Ring.
He tries to keep living with his family, but Sméagol has developed the very bad habit of putting on the Ring to disappear and then playing tricks, spying, or stealing from his relatives. So his grandmother boots Sméagol out of their home.
Around 2,470 of the Third Age (see Appendix B of The Return of the King for this date), Sméagol seeks refuge in the caverns under the Misty Mountains.
He hangs out under the Misty Mountains for about five hundred years, slowly turning into Gollum.
Bilbo Baggins turns up under the Misty Mountains on his adventure to kill Smaug the Dragon, and finds Gollum's Ring on the ground.
The Ring has slipped off Gollum's finger, looking for a new master.
Gollum and Bilbo play a riddling game: if Bilbo loses, Gollum can eat him; if Bilbo wins, Gollum has to show Bilbo the way out.
Bilbo wins, but Gollum has gotten hungry and angry.
Gollum does not notice that the Ring is gone until he goes to look for it so that he can sneak up and kill Bilbo.
Bilbo slips on the Ring and becomes invisible.
He follows Gollum as Gollum tries to chase him to the exit of the tunnels of the Misty Mountains.
Bilbo escapes Gollum and carries the Ring to the Shire.
Gollum goes searching for the Ring, but he is captured and taken to Sauron in Mordor.
There, Sauron tortures Gollum to reveal that the Ring is in the possession of someone named Baggins, in the Shire. This evidence leads the Black Riders to hunt Frodo.
Gandalf asks Aragorn to help him find and capture Gollum.
Aragorn tracks Gollum along the edge of the Dead Marshes, where he captures Gollum, and brings him to the Elves who live in Mirkwood.
In Mirkwood, Gandalf questions Gollum and manages to find out all of Gollum's history with the Ring.
At the Council of Elrond, Legolas tells Gandalf that Gollum has escaped the Elves of Mirkwood.
Gollum quietly follows the Fellowship through Moria and Lothlórien and down the River Anduin, until Frodo and Sam leave the Fellowship at Amon Hen.