Why did the ring affect Frodo so much and not Bilbo?

I think the two main reasons behind this are:

• That Sauron is slowly increasing his powers over the whole timeline of the Hobbit and the Lord of the Rings - the stories of which take place over an 80 year period.

• That Bilbo never receives the attention of Sauron in quite the same way that Frodo does.

Bilbo's 111th birthday party which starts the story of the Lord of the Rings, and where Bilbo gives the ring to Frodo, takes place about 60 years after the events of The Hobbit. A further 18 years elapses before Frodo starts to bear the ring towards Mordor to destroy it.

In the distant past, Sauron was defeated by Isildur and thought to be destroyed, however he is slowly rebuilding his power for literally hundreds of years under the guise of being The Necromancer of Dol Guldur. Sauron does not even openly declare himself until about 10 years after Bilbo finds the ring, and he believes the ring has been lost in the area he was defeated by Isildur.

So at the time that Bilbo owns the ring, Sauron is not as powerful, and he believes the ring is lost and is searching for it in the wrong area.

In Tolkien's writings, Bilbo's party is in S.R. (Shire Reckoning) 1401. It is not until some time between 1409 and 1417 that Gollum (venturing into Mordor seeking the ring himself) is captured by Sauron. Since Gollum is released only a year before the Nazgul actively start seeking for "Baggins" one can guess that it is likely that this happens towards the end of this time period.

So it is well after Bilbo is no longer the owner and active user of the ring that Sauron starts to seek him. Frodo obeys Gandalf's recommendations and does not use the ring in this period, and breaks this for the first time 'accidentally' in the Prancing Pony very briefly, and does not seem to suffer particularly from the experience.

It is when he is attacked by the Nazgul on Weathertop that he first starts to see things - and even then it is primarily the nearby Nazgul that can sense and see him - not Sauron yet from his tower hundreds of miles away.

Also Bilbo stays in the Shire with the ring whereas Frodo brings the ring ever closer to Sauron in the attempt to destroy it, so its possible the closer he comes to Mordor the more powerful and corruptive a hold it gains on Frodo.

Frodo also KNEW more of the ring than Bilbo (Gandalf told Frodo whereas with Bilbo he pretty much just kept an eye on the hobbit). That knowing would alter how Frodo would use the ring - whether he willed it or not. Additionally, Bilbo never sought to DESTROY the ring, thus the Ring had no reason to protect itself from Bilbo and therefore had less of a grip on him.

Also, in the very end of the third movie(I haven't read the books in a long time :\ ) when they are headed towards the sea to leave with the elves, Bilbo, who looks like his life force is held on by a string asks Frodo if he still has the ring and even says, "I would have liked to have held it one last time". I agree, the movie timeline is really not apparent. You assume the extreme aging of Bilbo is because he no longer has the ring. The movies make it feel like the trilogy took half a year lol. Except for the end when Frodo says to Sam, "It's been four years...".

Why would Sauron ever think that the ring was destroyed? Is he not DIRECTLY linked to the ring? IF it was destroyed would he not also be killed?

There are a few reasons.

When Bilbo had the Ring Sauron was relatively inactive and geographically distant. It is fairly strongly implied that the Ring began to 'wake up' in some sense as Sauron began to return to power.

Bilbo never really understood the full significance of the ring while it was in his possession and it's entirely possible that that knowledge, just in itself was a factor.

Another important factor is the would he received on Weathertop. It's explicitly stated that the shard of the Morgul Knife was in the process of turning him into a wraith and so it seems logical that this might well increase the hold that the ring had.It is certainly mentioned at several points that this injury has a lingering effect.

Also Frodo comes much closer, physically to the influence of Sauron. On Amon Hen he comes close to being discovered by the Eye which by now is actively searching for him and again it is explicitly explained that as he gets closer to Mordor and Mount Doom specifically he becomes increasingly aware of the influence of teh ring to the point where is feels like a crushing physical weight.

What spurred Gandalf to come to the Shire and to organize the Fellowship to try and destroy the Ring?

It was because Gollum, searching to regain the Ring, wandered close to Mordor and was captured and interrogated by Sauron.

The entire time Bilbo had the Ring, Sauron was quietly biding his time, and gathering his Ring-less strength, while the Ring languished, lost, in the deep caverns beneath mountains, with Gollum.

After Frodo got the Ring, a stronger Sauron became aware that the Ring was not lost, and began to actively seek the Ring, both his his servants (the Nazgul, Sauruman) and with his own consciousness.

He was actively using his powers, focused towards the Shire, to try and find the Ring, and the Ring was seeking him, as well. It makes sense that it would be more likely to draw his specific attention in the latter timeframe.

You can't really know exactly what effect it had on Bilbo though. You don't see it.

Also, Frodo did too become invisible when putting on the ring. In the scene on the hill top with the ruins, he puts it on to escape from the Nazgul. However, he gets stabbed nonetheless because the Nazgul can see him. After all, they're Saurons elite units :p