In the movies, when Frodo first spies Gollum stalking them in the Misty Mountains (Moria) he informs Gandalf who tells him a little of Gollum’s history.
How did they know Gollum knew where the ring was?
He suspected that Gollum’s ring was powerful. By catching Gollum and interrogating him, Gandalf began to suspect that the Bilbo’s ring might be the One and discovered where Gollum had found it. He went to Minas Tirith because the last he knew of the One was that Isildur took it.
