When Bilbo falls into Gollum's cave in The Hobbit and he finds the ring on the ground, why doesn't he immediately put it on? Is there any textual evidence suggesting he was scared or nervous of putting it on or did Tolkien make Bilbo not put the ring on so he could have his whole conversation/introduction with Gollum?
Bilbo picked up the ring, pocketed it, and forgot about it. He didn't really even spare a moment's thought to it. Quoting directly from the Hobbit:
[Picking up the ring] was a turning point in his career, but he did not know it. He put the ring in his pocket almost without thinking; certainly it did not seem of any particular use at the moment.
He just had no reason to put it on, and was involved with some other stuff at the time. Bilbo even regards it as so insignificant a point that he forgets he has it at all (emph. mine):
"Ask us! ask us!" said Gollum. Bilbo pinched himself and slapped himself; he gripped on his little sword; he even felt in his pocket with his other hand. There he found the ring he had picked up in the passage and forgotten about.
"What have I got in my pocket?" he said aloud. He was talking to himself, but Gollum thought it was a riddle, and he was frightfully upset.
I think this is a good question, since we see from the rest of the story (in the Lord of the Rings) that the Ring meant to come to Bilbo. We read for instance in The Fellowship of the Ring, Gandalf says,
So now, when its master was awake once more and sending out his dark thought from Mirkwood, it abandoned Gollum. Only to be picked up by the most unlikely person imaginable: Bilbo from the Shire!
'Behind that there was something else at work, beyond any design of the Ring-maker. I can put it no plainer than by saying that Bilbo was meant to find the Ring, and not by its maker. [...]'
Why didn't he put it on right away? As @Emrakul has pointed out, there was no reason to put it on, but in a broader sense, it should be pointed out that Sauron was still far from being as powerful as he would be, and did not yet know that The Ring had been found. Therefore, his thought was not yet upon it, and he was not attempting to draw it to himself.
Because of this, as @Emrakul said, this Ring was still kind of just a sort of trinket to him, and though the Ring desired to be found, it itself had a lesser power than it would have. If you remember from The Return of the King, the Ring gained power as soon as it entered Mordor. Here, hidden underneath a mountain with its Master unaware of its presence, it had a lesser power: not yet enough to make its Bearer desire to put it on.