I hate to say this (and I SWEAR it's not a pun) but it depends. For our daughter, pull-ups were more of a hassle than they were worth. She treated them like a diaper and was not interested in going into big-kid undies until we took them away and replaced them with training panties (the thick kind that can be soiled without causing a level 4 hazmat incident). At that point, she took things seriously. We used them at night until she was ready to night-train, to slightly better effect, but still, not a good deal for her.
Now, for my nephew, pull-ups were the key. He was dragging his feet about training but the idea of being in big-boy pull-ups was intriguing, so he took them and then sailed right on to underpants.
So are they worthwhile? Depends on the kid and their motivation to train. Is there a right time to switch to them (or training pants of any other sort)? Let your child tell you; you'll know when they have gone from 'not sure if I need to go' to 'I need to go but I'm gonna ride this gravy train as far as I can.' When they get to the gravy train, time to go for training pants and drop the pull-ups. (I SWEAR these puns are not intentional!)
FWIW, I distinctly remember arguing with my parents around 18 months that I was old enough for big-girl panties. Mostly what I recall is the vast frustration that they just DIDN'T UNDERSTAND; after all, it was perfectly intelligible toddlerese. Neither of mine were NEAR ready at that stage. My son's pushing 30 months and still could care less.