However Bilbo didn't rapidly grow old, until the One Ring perished. He did get a bit older after he gave up the Ring; but it wasn't until the Ring was destroyed, that he began to show his real age of around 130— which had been reached by hobbits in Bilbo's family before.
Of the Ring-bearers, three were alive after the Ring's destruction, the hobbits Bilbo, Frodo, and Sam. Bilbo, having borne the Ring the longest, had his life much prolonged.
Bilbo doesn't significantly age until after the ring is destroyed. There is very little in the book that would indicate that he aged any quicker just because the ring was no longer in his possession.
While there is no official answer for why Bilbo and Sam were able to be rid of the Ring without too much issue, there is a lot of supporting evidence that suggests it was partially due to their inner morality, as well as partially because of the Ring choosing its own path.
Bilbo, too, would prove resistant to the Ring's influence, which has a bearing on why his apparent age is affected so much differently than Gollum's age. In comparison to Gollum, Bilbo has the Ring for only a short period -- decades instead of centuries -- which gives it far less time to work its will on him.
Gollum had the Ring for 500 years, which was long enough for its effects to be more or less permanent. Bilbo had the Ring for a number of years but far fewer years than Gollum. Gollum put the Ring on his finger only rarely, to catch fish. Because he did not wear it often, he never “faded,” like the Ring-wraiths.
Gollum had his life largely extended because he possessed the One Ring.
Hobbits are naturally more resistant to the influences of the ring than other races (Gandalf comments on this). Sam had only been carrying the ring for a short time, the longer the ring is in someone's possession, the more addicted they become to it. Sam's love for Frodo was too great for the ring to corrupt.
Hobbits are not totally immune to the Ring's effects, however, as shown by the changes it works in Frodo, Bilbo and Gollum. On the other hand, Boromir becomes obsessed with the Ring, but never possesses it, while Sméagol kills his kin Déagol, the first Ring bearer after Isildur, to obtain it.
Sauron infused it with a great deal of his power, so it is the power's corrosive influence that makes the ring dangerous to possess. Remember, even Gandalf said that he dared not take the ring (when Frodo offered it to him for safekeeping) lest it take his desire to help others and corrupt it.
That's easy. Gollum survived because he was needed by the author to cause the ring to fall into the crack of doom to it's destruction (and to antagonize and guide Frodo and Sam, and show Frodo and Faramir's mercy, and Bilbo's pity, and to prove that even the wisest cannot see all ends).
In the movies Sam never wore the Ring, but in the books he did after carrying it 2 days.
Frodo came of age as Bilbo left the Shire. Frodo inherited Bag End and Bilbo's ring. Gandalf, uncertain about the origin of the ring, warned Frodo to avoid using it and to keep it secret. Frodo kept it hidden for the next seventeen years, and it gave him the same longevity it had given Bilbo.
In the spiritual world, beings are invisible to those in the physical world. This is why Isildur, Bilbo, and Frodo become invisible when they wear the Ring—because they are no longer in the physical world. As Olsen said, Sauron does not have a real body, so he is not affected in the same way.
He had already celebrated his 131st birthday, becoming the oldest Hobbit in the history of Middle-earth. As a mortal, he died in the West. While sailing west, Bilbo composed a last poem looking back on Middle-earth in farewell.
The One Ring may affect Frodo more than Bilbo, but it still corrupts Bilbo to a certain extent. The Ring's influence on Bilbo is clear when Gandalf has to force him to leave it behind in The Fellowship of the Ring. Bilbo has clearly grown attached to the Ring after all the years he has been in possession of it.
Frodo asks Gandalf to take the Ring, but the wizard refuses vehemently. With the Ring, Gandalf says, he would become too powerful, and he would inevitably be corrupted like Sauron himself. Even if Gandalf took the Ring simply for safekeeping, the temptation to use it would be too great.
The Ring cannot effect Tom Bombadil because he is outside the whole issue of Power and Domination; Tolkien uses Tom as an allegory that even this intense struggle between "good and evil" is only part of the whole picture of existence. On this page you can find everything about the Rings of Power.
After all, Sam's loyalty is partly a result of his servant status. Since Sam knows his place and has no ambition to rise socially, he never appears to envy Frodo's wealth or authority in the Shire. In fact, it's quite the opposite: Sam loves and admires his master, as though Frodo's honor reflects well on Sam himself.
Hobbits are simple creatures and do not wish for much, so there is nothing for the Ring to use against a hobbit. Frodo is the hobbit that is corrupted the most next to Gollum and Bilbo, which is why he is unable to throw the Ring into the fire. But even Frodo remained good at his core.
Merry and Pippin may presumably have also been very good at resisting the ring, as they are hobbits, and neither seem to have selfish ambitions nor lofty desires, so it's safe to assume that they would have passed the test unscathed.
Because of the power of the Ring (which he inherited from Bilbo on his 33rd birthday), Frodo's physical aging appeared to stop (at least outwardly).
Poor wretch! He has done us no harm.” And at this point Frodo remembers the conversation with Gandalf, when Frodo said he had no pity for Gollum, and says that “For now that I see him, I do pity him”. Gollum promises to serve Frodo, and swears by the Ring.
Though Gollum was certainly deeply affected. Yet it would not have been in the Ring's interests, or Sauron's either, to turn Gollum into a wraith. Firstly, he would not be much use in battle. And second, he wanted the Ring as much as Sauron did.