DVK's answer to this question states that invisibility is the trait Hobbits get when wearing the One Ring, because this is an enhancement of their already stealthy nature. He also suggests that other races received different augments based on their racial qualities.
What effect does wearing the One Ring have on humans?
Isildur became invisible too. From Unfinished Tales, the Disaster of the Gladden Fields chapter:
Isildur turned west, and drawing up the Ring that hung in a wallet from a fine chain about his neck, he set it upon his finger with a cry of pain, and was never seen again by any eye upon Middle-earth. But the Elendilmir of the West could not be quenched, and suddenly it blazed forth red and wrathful as a burning star. Men and Orcs gave way in fear; and Isildur, drawing a hood over his head, vanished into the night.
From this we can tell Isildur became invisible ("was never seen again by any eye upon Middle-earth") but in contrast the Elendilmir was still visible, until Isildur's hood was pulled over it. This then explains why Isildur was spotted and shot by Orc archers once the Ring had slipped off his finger. Interestingly, the only real difference between UT and the movie is that while Isildur was spotted swimming in the movie, he was shot while leaving the river in UT.
For those, like me, who didn't know what the Elendilmir is, it was a crown of the Kings of Arnor, containing a white crystal of Elven origin.
Gandalf also implies that all Men (and Hobbits) would become invisible, saying in LotR:
A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings.
Since Hobbits are (to my best guesses, it's never really established in canon but then their origin with regards to creation is not revealed at all) most likely an offshoot of the Race of Men, it makes sense that they react the same way to the One Ring. There is no way to know what would happen if an Elf or Dwarf were to put on the One Ring, though it might be assumed that should a Maia like Gandalf or Saruman were to wear the Ring they would become a great deal more powerful than they already are.
Invisibility was an effect that ticked on with mortals. With one exception, the Dwarves. The Rings were meant to enslave the free people, but they did not work on the Dwarves.
And all those rings that he governed he perverted, the more easily since he had a part in their making, and they were accursed, and they betrayed in the end all those that used them. The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows." [Silmarillion; Of the Rings of Power and the third Age]
The Rings increased the lust for gold in Dwarves. Gandalf tells Frodo;
A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every moment is weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings. Yes, sooner or later - later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last - sooner or later the dark power will devour him! [The Shadow of the Past]
With Men and Dwarves the effects are different because the effects did not work on the Dwarves due to their nature to resist the dominion of others. The One Ring itself is a different matter as Gandalf elucidates:
so powerful that in the end it would utterly overcome anyone of mortal race who possessed it. It would possess him.
Isildur himself being mortal would also be granted the abilities the Ring/s gave to mortals, "when he wore it he was invisible to all eyes" [Of the Rings of Power and the Thrid Age]. When he last wore it "he set it upon his finger with a cry of pain, and was never seen again by any eye upon Middle-earth." [The Disaster of the Gladden Fields]
