You mentioned animosity between
Gandalf and
Saruman. As far as I know, there wasn't any until Saruman openly displaced his treachery.
Merroe is correct. In an essay titled “The
Istari” in
Unfinished Tales edited by Christopher
Tolkien along with much commentary on his father’s notes, there is considerable animosity between
Sauron and Gandalf, beginning before
Bilbo’s journey to the
Lonely Mountain with the
Dwarves. Most of this animosity was on Saruman’s part, fueled by his jealousy of Gandalf; but some of it was on Gandalf’s part as he reacted to Saruman’s sarcasm, belittling, and obstruction.
Saruman deduced that
Círdan had entrusted
Narya,
the Ring of Fire, to Gandalf upon his arrival in
Middle-earth and was jealous of him for that. Soon after he settled in
Isengard, he began searching for
the One Ring in order to wield it himself. Gandalf on his part might have deduced Saruman’s envy of Gandalf’s position as keeper of one of the Three Rings; but he could not know Saruman’s ulterior motive: his coveting the One Ring. Gandalf’s distrust of Saruman can be interpreted as a premonition:
just luck, as Tolkien occasionally puts into the mouths of characters when they tongue-in-cheek reference the intervention of a Higher Power in Middle-earth.
As for an interest in
the Shire, I don't see why that would make Saruman bad, at least as far as Gandalf knew. More to the point, we don't ever know HOW Saruman knew that
the Ring was even there. I did, however, hear that it was
Grima Wormtongue who told the
Nazgul WHERE the Shire was, so presumably Saruman knew at some point.
It was the squint-eyed Southerner at
The Prancing Pony in
Bree who gave up
maps to the Shire and the names and locations of important people there when the
Nazgûl captured him near Tharbad. (See
Reader’s Companion.) He was Saruman’s agent, but betrayed Saruman to the
Witch-king partly out of sheer terror of the
Ringwraiths and in order to save his own life.
As for Saruman’s assessment that the Ring was in the Shire and his (correct) guess that Gandalf knew where it was bestowed, he deduced that from the movements of the Nazgûl and the fact that they were asking for the location of “
Shire” reported to him by
Radagast.
Saruman paid visits to the Shire in disguise because he wanted to know what Gandalf was doing. He also took up smoking pipeweed, a habit for which he ridiculed
the grey wizard. Gandalf learned of these visits, but merely laughed them off at the time, assigning no wickedness to Saruman.
By the way, several people knew about Bilbo’s Ring by the end of
The Hobbit. The surviving members of
Thorin & Co. knew, because Bilbo told them. (He deceived them about how he obtained it, however, telling them
Gollum gave it to him for winning the
Riddle Game.) Gandalf knew, and you are likely correct,
BalrogRingDestroyer, in guessing that the Dwarves told him; but for that we cannot be certain. The
Elvenking probably knew, too, because he told Bilbo at their parting, “May your shadow never grow less (or stealing would be too easy)!” It is possible Gandalf told him about the Ring; but the Elvenking was no dummy: he was wise and (in Hobbit terms) very ancient, and had surely investigated the escape of Thorin & Co. from his forest
fortress most thoroughly, enough to determine that there had been an invisible thief on the premises. Finally, Gandalf certainly told
Elrond at some point: we just don’t know if he did that at their first meeting on his return with Bilbo or later.