Some thoughts on why Gandalf takes so long to figure out that Bilbo's ring is the One Ring.
1. It is important to remember that the One Ring is a danger, if re-possessed by Sauron, but not a potential salvation. At the time of the Council of Elrond, there is no assumption that destroying the Ring will destroy Sauron and produce victory. It will just deny Sauron the Ring. So, even if the Ring still exists in Middle-earth, as long as Sauron does not get his hands on it, it is not a threat. (Or, is that true? At some point it becomes clear to Gandalf that the Ring might (will) become a threat if anyone powerful enough gets possession of it. By the time of 'The Shadow of the Past' Gandalf will not take the Ring, suspecting that the Ring could 'gain a power' over him, and he might 'become like the Dark Lord himself'. This is why Gandalf favors destroying the Ring, rather than hiding it or throwing it in the Sea during the Council. But when does Gandalf form this theory, and why?)
2. So, the Shire is a pretty safe place for any ring, even The Ring. No one in the Shire is powerful enough to wield the Ring and become a new Dark Lord. The Shire is protected in the depths of Eriador, by Rivendell guarding the East, Lothlorien guarding the Redhorn pass, Saruman guarding the South, Cirdan guarding the West, Gondor providing a defence against a break-out from Mordor, the Dunedain providing a close guard, by secrecy and the lack of knowledge about the Shire, and by Bilbo, who is a pretty wise, experienced and redoubtable Hobbit. The Shire is perhaps the safest place in Middle Earth to store any dubious rings, so, that reduces pressure to do anything quickly. Whatever that ring of Bilbo's is, it couldn't be in a safer place.
3. As discussed in class, the assumption is that the One Ring has been removed from the playing field by the Valar.
These are perhaps the main reasons why Gandalf shows a lack of urgency in questioning the nature of Bilbo's ring.
Let's look again at the time-line:
2850: Gandalf enters Dol Guldur and discovers that Sauron has risen again. 2851: The White Council meets, considers an attack on Dol Guldur, but decides against it on the council of Saruman. 2941: White Council meets again, attacks Dol Guldur. Sauron withdraws. Bilbo finds Ring under Misty Mountains. 2951: Sauron declares himself openly in Mordor, and begins re-building Barad-Dur. 2953: Last meeting of the White Council. Saruman states that the Ring 'was rolled down the river to the Sea', long ago while Sauron slept. 2954: Mt. Doom bursts into flame again. 3001: Bilbo's farewell party. Frodo becomes Ring-bearer. Gandalf suspects that Frodo's ring might be the One. 3009: Gandalf and Aragorn renew the hunt for Golum. 3017 Gollum is captured by Aragorn. Gandalf reads Isildur's scroll. 3018, April 12: Gandalf reaches Hobbiton, and the next day puts the ring in the fire and determines that it is The Ring.
So, some thoughts on the timeline. If Gandalf had questioned Bilbo's ring when Bilbo first got it, and decided that whether it was the One Ring or not, it was best destroyed in Mt. Doom, the ten years between 2941 and 2951 might have been the time to do it, as Sauron was not yet in Mordor, so the way to Mt. Doom might not have been well defended? But wait! Would that have worked? Mt. Doom had not yet burst into flame again. Can you destroy a ring in a dormant volcano? Once Mt. Doom had become an active furnace, Sauron had been embedded in Mordor for 3 years, building up the defences.
I imagine a conversation between different sections of Gandalf's sub-conscious, which may have repeated several times over the 67 years between Bilbo finding the Ring, and Gandalf knowing that it was The Ring. "I feel uneasy about that ring of Bilbo's." "So? Even if there was any reason to feel uneasy, what could you do about it that would be better than what is? It is secret. It is safe. It is in the safest place in Middle-earth. You have plenty of other more urgent things to worry about!"
More puzzling is Gandalf's lack of urgency once his suspicions grow, after Bibo's farewell party, and then after he knows for sure in April 3018.
However, again, why is there urgency? Remember, that Gandalf does not know that destroying the Ring destroys Sauron. He only knows that Sauron getting the Ring would be a disaster, as Sauron would become irresistible. So, is he going to do anything different if Bilbo's ring turns out to be the One Ring? Keeping it secret and safe in the Shire is a lot less risky than trying to sneak it into Mordor to destroy it. He thinks that The One Ring has been removed from the playing field by the Valar. He probably thinks that Sauron thinks that the One Ring has been removed from the playing field by the Valar (one reason he retreated from Dol-Guldur to Mordor so readily). After Bilbo's party, Gandalf's vague misgivings become active concerns, but one reason that urgency seems to be lacking is that there would be no alternative plan of action if Bilbo's ring happened to be The Ring. The only reason to do anything besides keep that ring secret and safe in the Shire is if Sauron somehow becomes aware that it still exists in Middle-earth, and that it is in the Shire.
Now, if Gandalf thought that destroying The Ring would destroy Sauron, then he might have had a greater sense of urgency. "If I could only find Sauron's Ring, and somehow get it safely cast into Mt. Doom! Sauron would go poof! Mission accomplished! Back to a cosy life in Valinor for me!" But, Gandalf did not have that knowledge, nor that assumption, until sometime after the Council of Elrond.
Gandalf's main fault, I think, comes once he is aware, (generally from Gollum, and urgently from Radagast) that Sauron is aware, and the Nine are on the move. He should not have left Frodo, to go chasing off to Isengard. .Gandalf has avoided discussing Bilbo's ring with Saruman so far, but his reaction of relief and his dash to Isengard on hearing Saruman's invitation relayed by Radagast, strikes me as a bit of hasty panic, upon hearing that the Nine are in Eriador. Collecting Frodo, and hotfooting it out of the Shire, perhaps to hand Frodo off to Aragorn before heading for Isengard, would seem a more thoughtful plan.
In any event, there are some questions in this whole affair? How does Gandalf form the theory that wielding the Ring turns someone into a new Dark Lord. There is no historical evidence that this is so. So Gandalf's possible sources of this theory could be: the studies and speculations of Saruman; his own experience of his own ring; theories and speculations from Elrond, that might have derived from Celebrimbor's ring makers; info from The West?
How and when does Gandalf come to know (or guess) that destroying the Ring will destroy Sauron? He states this at the Meeting of the Captains of the West, after the Battle of the Pelennor Fields. He didn't have this assumption at the Council of Elrond. Did this knowledge somehow result from his transformation to Gandalf the White?
To summarize: I think that Gandalf's lack of urgency is generally caused by the fact that as long as Bilbo's ring is secret and safe in the Shire, this is the best possible place for it in Middle -earth. Sure, destroyed, in Mt. Doom would be even safer, but getting there any time after Mt. Doom is an active volcano again is almost insanely risky. It is only after The Ring and The Shire are no longer secret, and no longer safe, that alternative plans have to be urgently considered. This, first lulls Gandalf's misgivings, then, once his vague misgivings have become active concerns, still makes addressing those concerns not particularly urgent.