[The Might]

The Ring's power was definitely not to turn a person invisible, as it would have made little sense to do this. Why would Sauron had given it this power, if he himself was visible while wearing it?
My suspicion is that its power was to transport a person from the Seen into the Unseen World, also called Wraith World, but the reason for this I could only speculate, perhaps that Sauron could thus better control or communicate with the Ring-wraiths.

About other wearers...most would be turned invisible.
But certain beings existed in both worlds. Ringwraiths had a more material look in the world and High Elves (those that had journeyed to Valinor and seen the light of the trees + Elu Thingol) appeared as a shining figure.
It could be that they would not disappear, as they already existed in the Wraith-world.

As for Bombadil, as he is not of Middle-earth he is not really bound to the laws that exist in the world but only to his own laws. As such the Ring had no power over him and also he could see Frodo while he was wearing it.