How did Gandalf and Mordor come to know about Gollum?

Question

It's been about a decade since I've read The Hobbit or The Lord of the Rings, but I recently re-watched the movies (all 6) and from the standpoint of the movie, there is no clear indication that Gandalf ever found out about Gollum and his relation to the One Ring. In the beginning of The Fellowship of the Ring, Gandalf remarks that the ring has been called "precious" by Gollum before, after Bilbo calls it such. He also indicates to Frodo that he "searched everywhere, but the enemy found [Gollum] first". I ask this question because, in The Desolation of Smaug, Bilbo never explains how he found the ring, much less what it does. Gandalf, of course, is suspicious and (Battle of the Five Armies spoilers)

By the end of the 3rd movie, he indicates that he knows Bilbo has a magic ring, but he does not realize at that moment which ring in particular it is

Frodo knows of Gollum too, but I assume only because he lived with Bilbo and likely heard stories about his adventures. Gandalf hasn't seen Bilbo since the events of The Hobbit (some 60 years I think), so there's no way he would've learned about Gollum from Bilbo in-between The Hobbit and The Lord of the Rings. I assumed, perhaps, that Gandalf somehow learned of Gollum during his research in Minas Terith.

So how and when does Gandalf actually learn about Gollum? And how did Mordor/Sauron learn of Gollum to capture him to find information about the whereabouts of the Ring?

Answer 1

In the book Gandalf knows about Gollum because Bilbo told the company - although he withheld information about the Ring at first:

Then they wanted to know all about his adventures after they had lost him, and he sat down and told them everything-except about the finding of the ring ("not just now" he thought). They were particularly interested in the riddle-competition, and shuddered most appreciatively at his description of Gollum. (Out of the Frying-pan, Into the Fire)

Eventually after the Spiders episode (Flies and Spiders) Bilbo tells the Dwarves everything (it should be noted that Gandalf is not part of the company at this time):

There they lay for some time, puffing and panting. put very soon they began to ask questions. They had to have the whole vanishing business carefully explained, and the finding of the ring interested them so much that for a while they forgot their own troubles. Balin in particular insisted on having the Gollum story, riddles and all, told all over again, with the ring in its proper place.

I can't at present source a reference in The Hobbit for when Bilbo told Gandalf about his Ring (although Gandalf did know about Gollum as my first quote shows) but Gandalf evidently did know, and knew quite early, as is shown in The Shadow of the Past in Lord of the Rings:

Then I heard Bilbo's strange story of how he had "won" it, and I could not believe it. When I at last got the truth out of him, I saw at once that he had been trying to put his claim to the ring beyond doubt.

As for Sauron, there's absolutely no evidence that Sauron knew anything about Gollum and Gollum's possession of the Ring before Gollum's entry into Mordor, but after that he did find out quickly enough; again from Shadow of the Past:

'But I am afraid there is no possible doubt: he had made his slow, sneaking way, step by step, mile by mile, south, down at last to the Land of Mordor' .....

'Wretched fool! In that land he would learn much, too much for his comfort. And sooner or later as he lurked and pried on the borders he would be caught, and taken – for examination. That was the way of it, I fear. When he was found he had already been there long, and was on his way back. On some errand of mischief. But that does not matter much now. His worst mischief was done.

'Yes, alas! through him the Enemy has learned that the One has been found again. He knows where Isildur fell. He knows where Gollum found his ring. He knows that it is a Great Ring, for it gave long life. He knows that it is not one of the Three, for they have never been lost, and they endure no evil. He knows that it is not one of the Seven, or the Nine, for they are accounted for. He knows that it is the One. And he has at last heard, I think, of hobbits and the Shire.'

So yes, Gollum did simply walk into Mordor.

Answer 2

His research in Minas Tirith was rather the other way round. As the other answers have said, Gandalf knew about Gollum from Bilbo. 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. He was trying to tie Gollum's finding to the last known location of the One Ring and succeeded.

Answer 3

In the op you mentioned gandalf saying it had been called precious by gollum. That is false. He said it has been called that before, but not by you. He had no way of knowing that gollum called it precious, he had read of Isildur saying it was precious to him.