Greentext

“Yreve emit ew tih mih htiw eht lugrom edalb, eh setalucaje. Eh seunitnoc gniggeb rof erom. Enon fo su era elbatrofmoc gnihcuot mih evila ro daed. M’I yrros ym drol.”

pænus

Well damn. This guy Middle Earths. Thanks for the lore drop!

I've always wanted to see what would have happened if Gandalf or Galadriel took the ring

It does, in fact, turn humans invisible too. Isildur being the obvious example, but even the nine rings given to humans had that effect, shifting them to the spiritual / unseen world. That's a whole different ramble, but for now, let's sum it up that there is an unseen world not everyone can sense and influence, but the Maiar (including Sauron) are inherently spiritual beings that took physical shape in the seen world in order to interact with it.

For Sauron, so much of his power was poured into the One Ring that the continued existence of the ring meant that he could survive destruction of his physical form and eventually take shape again¹, though its loss in the War of the Last Alliance obviously robbed him of much of his strength and he had to hide for a long time to slowly regain his strength and renew his efforts.²

Through the Ring, Sauron had also dominated the nine human Ring bearers and bound them to him, moving them into the spiritual world. As his form was destroyed, so did they lose theirs. As he returned, so did they.

The reason they could still "see" Frodo is that they were attuned to the unseen and could sense him there, with their power over it manifesting in them stabbing his physical form even though it was invisible to mortal eyes.

There is still the question of the Dwarven rings. They were forged first, and it's possible they weren't as refined yet, though the dwarves are also described as more resilient at resisting the dominating effect. My guess is that the fact they were created by Aulë, Smith of the Valar, rendered them less susceptible to the craft of a lesser spirit (Sauron), but I have no evidence.

1: This paragraph and the following one originally read that Sauron could no longer take shape without the ring. I stand corrected on that, see the responses.

2: After his first destruction during the Fall of Numenor, his spirit managed to escape with the ring. 110 years later, he had enough strength to launch a strike against the nascent Gondor and start another war that lasted 13 years. He was destroyed again, this time losing his ring, and it took him a thousand years to become active again.Per the correction, he will probably have regained his form, though he was still too weak to fully reveal himself and start another open war for two thousand more years.

It wouldn't turn a human invisible because it only turns hobbitis invisible.

See: the intro to the first movie. Isildur, after his victory at Mt Doom, uses the Ring to turn invisible in attempts to escape a group of orcs that attacked his party.

The one ring gives the wearer the power of what the wearer defines as power. Frodo turns invisible because for a hobbit not being noticed means safety. The ring tells Sam that he could plant as many Apple trees as he likes, even plastering whole middle earth.

In this moment the ring probably tells to Boromir that he could rebuild Gondor to old strength and he might be able to (before giving in to saurons influence).

Invisibility is incidental; it's a side effect for those who do not exists in the unseen realm. It was created to dominate those who had the lesser rings of power; the dwarves proved too resilient, and the elves took theirs off before they could be influenced. The ring is a tool of domination, and enhances the bearer's natural abilities. It may have allowed Aragorn for example, to inspire people to follow him as he became the next great dictator. It is a more subtle magic. In the books, Gandalf describes this process if he were to claim the ring; that he would build up a great kingdom to overthrow Sauron militarily and replace him as Middle Earth's tyrant.

Generally it will work on any mortal except dwarves.