Quoting the same passage as Maédhros used:
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"At the worst," said he, "our Enemy knows that we have it not and that it still is lost. But what was lost may yet be found, he thinks. Fear not! His hope will cheat him. Have I not earnestly studied this matter? Into Anduin the Great it fell; and long ago, while Sauron slept, it was rolled down the River to the Sea. There let it lie until the End."'
Gandalf fell silent, gazing eastward from the porch to the far peaks of the Misty Mountains, at whose great roots the peril of the world had so long lain hidden. He sighed.
`There I was at fault,' he said. `I was lulled by the words of Saruman the Wise; but I should have sought for the truth sooner, and our peril would now be less.'
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Saruman was, after all, Gandalf's superior, and had studied the rings a lot. Gandalf had at that time no reason to mistrust Saruman.
Quote:
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"The Nine, the Seven, and the Three," he said, "had each their proper gem. Not so the One. It was round and unadorned, as it were one of the lesser rings; but its maker set marks upon it that the skilled, maybe, could still see and read."
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He could immidiately dismiss the thought of it being one of the Nine, Seven or Three. And what would the chances be of a hobbit picking up the long lost One Ring? The chances of him finding a "lesser" ring would probably be bigger.
I think Gandalf did what he could. After TA 3001 when he picked up his suspicions, he went around hunting Gollum for a while as well. And he might also have gone to other places to search, mainly Rivendell, like someone said.