Gollum came to appreciate the Ring's power over him.  He started out with envy and murder in his heart, and started his ownership with it with the murder of Deagol.  As has been pointed out, the Ring took his weaknesses and played on them.  With the Ring he became a serial murderer, as even killing Orcs is murder in the manner he killed them.

Bilbo found the Ring, and didn't purposely take it just to possess it.  He started his ownership with it with mercy instead of evil actions, and used its power only to assist others.  He was not a great enough person to try to do more good than that necessary for himself and his companions, so it was harder for the Ring to ensnare him through personal pride or ambition.  And once home, the only time he used it was to escape the attentions of the Sackville-Bagginses.  He never realized what the Ring was, where it came from, or what its discovery meant to the world of Middle Earth.  His very ignorance of what the Ring was, its power and its nature, also served to protect him.

Frodo, on the other hand, learned the nature of the Ring and had far more appreciation for the evil it represented, particularly as he saw two of the greats of his day actively deny it (Gandalf and Galadriel) and two others (Elrond and Aragorn) from the first seeing it needed to be destroyed.  He also came to this knowledge after the Ring had begun to awaken, after Sauron had begun to recover his powers and to call for it to return to him.  He carried it the furthest with the most knowledge of what it could do to him and to others.  He had seen Gandalf's naked fear of what the Ring would do through him if he kept it, and saw Galadriel's appreciation for what she would become.  And he heard Boromir's plans for what he would do with the might of the Ring at his command, and had an appreciation from what Gandalf and Galadriel told him of how this was a trick of the Ring's--to ensnare a new bearer.

For Frodo, at the last, there in the seat of its power, at the last it overwhelmed even him.  He'd been able to withstand it for the longest time, partly because he was a hobbit, and hobbits aren't particularly ambitious folk to start off with.  He was a gentle soul, and gentleness wasn't something the Ring could catch hold of easily, either.  He was compassionate, and that was another trait the Ring would have a hard time using, although Gandalf indicated that would be the strategy the Ring was most likely to use in ensnaring himself.  He never killed along the way, which left him without that kind of guilt to use in corrupting his soul.

I was glad, however, in seeing the films, to recognize that even at the end PJ and/or Elijah let that one short glimpse of horror at claiming the Ring be expressed by Frodo before he returned to the sneering smile worn by Isildur when HE claimed the Ring.  There was no hint of that in the books, of course, but to see it there in the film was in some way reassuring.

For, in the end, it wasn't truly the individual who claimed the Ring who was doing the claiming--it was the Ring claiming another victim instead.

"Then do not be too eager to deal out death in judgement. For even the very wise cannot see all ends."

"Don't go where I can't follow!"