Gollum finds out Bilbo's name and where he is from when they meet in the cave.
Why doesn't he ever go to the Shire to search for Bilbo and his "precious"?
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Smeagol was a Stoor Hobbit before the ring corrupted him. This means he was from Gladden Fields just east of the Misty Mountains.
According to the link below he did go searching for the ring but he waited until 2 years after Bilbo took the ring. He spent almost the next 60 years either searching or being captured.
I reckon he never went to the Shire because he didn't know where it was. Shire Hobbits didn't seem knowledgable about things outside of the Shire so I imagine Stoor Hobbits would be the same. I'd say its safe to assume that Smeagol was never west of the Misty Mountains so he'd know nothing of that land. Also I can't see Gollum asking someone for directions to the Shire so he'd have no way of knowing where it is.
The green circle is Galdden Fields and the red circle is the Shire.
The Fellowship of the Ring contains a description from Gandalf about his thoughts about what happens after Bilbo takes the ring.
Gollum is afraid of the light after so many years living under the misty mountains so it takes some time before he leaves them to search for the ring. Gandalf then surmises that, because of his very long association with the ring, he is drawn by Sauron's growing power and makes his way south to Mordor and is eventually captured and tortured.
All Gollum knows is that the ring was taken by 'Bagginses' from 'The Shire', but from his actions clearly doesn't know where that is.