Bilbo, himself, first meets Gollum after he is separated from Gandalf and the dwarves during a goblin attack. Lost, tired and scared, he stumbles upon a gold ring in the dark passages, pockets it, and thinks no more of it.
Eventually, he finds his way to a lake, which just happens to be where Gollum lives. Since he has just recently eaten, Gollum simply observes Bilbo, before speaking to him, wary of the sword in Bilbo’s hand (later named Sting).
It may be that Gollum has become lonely in his long years under the Misty Mountains, and feels a connection with Bilbo, being hobbit-like himself, but for whatever reason, he convinces Bilbo to have a riddle competition (see Riddles in the Dark).
When Bilbo is finally stumped for a riddle, he asks aloud, “What have I got in my pocket?”, which Gollum mistakes for a riddle. After three wrong guesses, Gollum is forced to fulfil his promise to lead Bilbo out of the mountains. But seeing as he never planned on doing this (being quite confident in his riddling skills), he returns to his little island to get his “precious” – a magic ring that allows him to become invisible – only to find it missing. He puts two and two together and works out what we already know: Bilbo has “the precious”.
Unaware of the power of the Ring, Bilbo accidentally slips it onto his finger, and thus, Gollum, who is about to kill Bilbo, runs straight past him. Gollum, in a fit of rage and despair, unwittingly leads Bilbo to the exit, and allows Bilbo with the Ring to escape from the maze of passages within the Misty Mountains.
Gollum just misses catching Bilbo as he jumps over the wretched creature. Defeated – as he dares not go any further for fear of goblins – Gollum cries out in pain from the dark, “Thief, thief, thief! Baggins! We hates it, we hates it, we hates it forever!”
