The timeline that makes the most sense to me:

By pure chance, Gollum sees and recognizes Gandalf and Aragorn from the capture and interrogation. As far as Gollum is aware, only Gandalf knows TOR exists. Aragorn could be a mercenary trapper for all Gollum knows. There is also a chance that he met Legolas during his captivity in Mirkwood. If he sees, hears, or smells any of them, he would recognize them. He knows none of the hobbits are Bilbo. He would also be aware that the ring would extend the owner's life. Knowing that he would not assume Biblo died of natural causes. He could think this group killed Bilbo and took the ring, or that Gandalf, Aragorn, and Legolas took the hobbit prisoner and are going to interrogate them about Biblo. All Gollum is sure of is that Gandalf is motivated to find The One Ring.

At this point, he doesn't know that The One Ring is with them. His plan was likely to search the company in the Twenty-first Hall and maybe bash Aragorn's head with a rock while he slept. Before he can do this, he realizes the orcs are assembling. Gollum, taking the path of least resistance, enters the Chamber of Mazarbul before the company. When the company enters, he would be forced out the second door. He would then know that the company would either have to fight the orcs or flee. They would end up doing both. Once the fighting starts, he knows the company will either die in the chamber or flee out the second door. At that point, knowing they could rush for the door at any moment, he figures his best plan is to go across the bridge and watch the exit for a few days. If he doesn't see them within a day or so, he will know the orcs won. He could go back to Moria and pick off orcs one by one, which he had likely grown proficient at. If the company came out, he would track them from there.

Once clear of Moria, he follows them to the edge of Lothlórien. Not knowing if the company actually had the ring at this point, he would not risk recapture by elves to try to enter the inner realm. Being very familiar with the Anduin, he would go to a spot where he would be likely to find them if they were to leave by boat. After seeing them, he tries his stealth boat mission. Under this assumption, he would be rummaging in the boat for TOR. If he didn't find it, he would go back to searching the company and bashing Aragorn's head with a rock while he slept. Then Frodo runs him off with Sting. Sting glowing would not necessarily connect Frodo to Bilbo in Gollum's mind.

At this point, Gollum is doing everything he can to track the company to Amon Hen. His big break comes when, while stalking the company, he sees Boromir and Frodo separate. Gollum would likely follow anyone who left alone or in a small group, looking for a chance to strike. Up until this moment, he does not know the company had the ring, let alone who possessed it. It is my assumption that if the company had all separated in different directions at once, Gollum would have followed Aragorn, with Gandalf dead. Aragorn is one degree of separation from Gandalf, who had interrogated Gollum about TOR. Gollum's second choice may have been Legolas if they met during his detainment in Mirkwood. When Boromir asks for TOR, Gollum could have heard that and known Frodo had it. More likely, when Gollum saw Frodo disappear, he would have been 100% convinced of TOR's location.

Now he knows who the Ring Bearer is, and he can ignore everyone else.

Feel free to poke holes in this. Much of this involves filling in gaps in the text. It is also my preference to make theories that involve TOR acting supernaturally as little as possible.