[Legolas]

Gandalf's ring was not a magic fire-blasting wand-type weapon or anything; nor were any of the rings that we know of. The power of the Three is described by Elrond at the Council of Elrond:

'The Three were not made by Sauron, nor did he ever touch them. But of them it is not permitted to speak. So much only in this hour of doubt I may now say. They are not idle. But there were not made as weapons of war or conquest: that is not their power. Those who made them did not desire strength or domination or hoarded wealth, but understanding, making, and healing, to preserve all things unstained.'

And in Appendix B, Cirdan tells exactly why he gives Narya to Gandalf:

For this is the Ring of Fire, and with it you may rekindle hearts in a world that grows chill.

Gandalf put Narya to use for the exact mission he was sent: to rouse the good people of Middle-earth to action, to inspire and rally them together.

Saruman did eventually know that Gandalf had a ring. In the Istari chapter in Unfinished Tales:

"But Cirdan from their first meeting at the Grey Havens divined in him the greatest spirit and the wisest; and he welcomed him with reverence, and he gave to his keeping the Third Ring, Narya the Red.

'For,' said he, 'great labours and perils lie before you, and lest your task should prove too great and wearisome, take this Ring for your aid and comfort. It was entrusted to me only to keep secret, and here upon the West-shores it is idle; but I deem that in days ere long to come it should be in nobler hands than mine, that may wield it for the kindling of all hearts to courage.'

And the Grey Messenger took the Ring, and kept it ever secret; yet the White Messenger (who was skilled to uncover all secrets) after a time became aware of this gift, and begrudged it, and it was the beginning of the hidden ill-will that he bore to the Grey, which afterwards became manifest.

As Pervinca Took alluded, tension was already present between Saruman (Curumo) and Gandalf (Olorin) from the moment they were chosen to sail to Middle-earth as emissaries. In the same chapter as above, there is essay about a meeting of the Valar where the wizards are comissioned; this detail is recorded:

Manwë replied that he wished Olorin to go as the third messenger to Middle-Earth. But Olorin declared that he was too weak for such a task, and that he feared Sauron. Then Manwë said that was all the more reason why he should go, and that he commanded Olorin (illegible words follow that seem to contain the word "third"). But at that Varda looked up and said: "Not as the third;" and Curumo remembered it.

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Originally Posted by Radtech51

So we know that it was Saruman’s task to learn everything he could about the rings of power, we also know that Saruman’s knowledge of the craft was very extensive. According to JRR Tolkien himself there existed a very real possibility of him finding the lost lore he needed and creating his own greater ring of power.

Knowledge of this ring craft wasn't specifically a 'task' assigned to Saruman, but rather it was a personal interest taken up by him in greed. Saruman would not have been in a great position to take Gandalf's ring, and he would have almost no use for it anyway. It might have even been risky.

Saruman, in his knowedge, would have know that the Three were never tainted, and accordingly did not submit to evil desires. His motivation was greed, thus Narya would've been useless to him. Gandalf tells Frodo as much in 'The Shadow of the Past':

He knows that it is not one of the Three, for they have not been lost, and they endure no evil.

Perhaps you might ask if he would want Narya simply to make sure Gandalf no longer had it. Though he did strand Gandalf atop Orthanc, there was no great fight (as the movie portrays...); to take Narya from Gandalf may have taken all of Saruman's strength, if indeed he could have beaten Gandalf at all. It was not worth it. At that point, Saruman's focus is to find the One Ring, which he suspects is abroad, and is the reason he confronts Gandalf to begin with. He knows time is of the essence, and suspects Gandalf knows where it is.

Even if he were to find a way to get Narya, Saruman's brash attitude suggests he probably wouldn't have kept ownership of it secret as the humble Gandalf did. Instead, he would have worn it proudly, drawing attention to himself. Sauron would've become aware of this, which wouldn't have been in Saruman's best interest.

It is worth noting that Saruman did manage to make his own ring, but it was apparently a failure. Hadn't worked out the kinks yet, I guess!
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Originally Posted by Radtech51

A: Did Gandalf, Galadriel or Elrond ever even use there rings of power to become invisible?
Surely we know that all 20 greater rings of power had this ability, so I’m going to assume that every one of the wielders except for the Hobbits knew how to control this power.

I've never read anything that suggests any of the rings aside from the One Ring had the ability to grant invisibility to its wearer. This is a trait unique to Sauron's ring.
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Originally Posted by Radtech51

Why didn’t the ring wraiths ever use there rings to become invisible in the books?

A: They most surely had in the beginning which is why the ring wraiths now walk in the twilight land permanently and are almost all but invisible to anyone looking upon them directly. I’m also going to assume that they could have used there rings in the story to make themselves completely invisible but didn’t see the need to do it.

In addition to the last question (the other rings didn't grant invisibility as far as I know), once dead, the Nine no longer wore their rings. Instead, Sauron kept them which is how he controlled their wills when he did not have the One Ring.

The Nine were present in the 'wraith-world' (what you call twilight) in death because they were mortals who wore the corrupted rings - not because of anything relating to invisibility. Though their bodies had died and faded because they were designed to be mortals, they were able to persist somewhat immortally through the corruption of the rings as long as the One Ring existed. Some other immortal beings have this trait (being present in the spiritual realm) as well.

We know at least High Elves could exist/were visible simultaneously in the physical realm and the spiritual realm. Gandalf explains to Frodo why he was able to see Glorfindel.

'And here in Rivendell there live still some of his chief foes: the Elven-wise, lords of the Eldar from beyond the furthest seas. They do not fear the Ringwraiths, for those who have dwelt in the Blessed Realm live at once in both worlds, and against both the Seen and the Unseen they have great power.'

'I thought that I saw a white figure that shone and did not grow dim like the others. Was that Glorfindel then?'

'Yes, you saw him for a moment as he is upon the other side: one of the mighty of the Firstborn. He is an Elf-lord of a house of princes.'