The first situation in which he used it to get himself out of a jam was when he was escaping from the cave in which he met smeagol, who we know is intensely aggressive when it comes to the ring. He slipped the ring on and it made him seemingly disappear. Another, much more trivial situation was when he bid everyone farewell at his birthday party basically as a show stopper (but also to avoid the judging eyes and boring personalities of his fellow hobbits).
The ring does not make Bilbo tougher, if anything, it makes him weaker. The power of the ring breaks him down mentally and physically.
