When do babies learn there own comforting skills?

Simple answer, when they need them!Yup, they don't need to learn to comfort themselves if they are always rocked to sleep, never let to cry, sleep in a swing or rocker the majority of time.
Babies care includes people meeting their needs by talking with them, feeding, burping, changing, and letting them sleep, alot. Their need for touch and connection are all met in those activities. Technically mother's of newborns can have plenty of time for work and sleep, if they let their children grow.
Playing is important too, but if your infant is fussy, you know they just ate, are not gassy, they are dry, have been up a while, but you think it is to early to sleep, don't try to cheer, don't say what's a matter, your just feeding the fussiness. Help them, put them down to hang out near you or put them to bed!! When you are tired and someone wants you to stay and visit and you get crabby, no cheering will help, you need sleep, same with baby.
So if we let them fuss a bit, they will not die. I found with all but one baby I have cared for over 20+ years, that children in general need 2 hour naps but will make some noise at the 1 hour point, but will go back to sleep. When parents tell me their newborn doesn't sleep well, I question it, Sometimes we need to leave them alone! They will go back to sleep if we haven't rushed in to rescue them. I don't sleep long if someone keeps checking on me :-)
Speaking of fussing when waking. Babies take their cues from us, if they fuss and in a sad voice, we say what's a matter, poor thing, you're feeding it. If they cry on waking, it's there communication but only as long as  you feed it, a cooing baby waking is so much more pleasant. When I get children up from naps, I always enter with a smile and positive comment, " oh I see a pretty baby that's awake, Hi sweetie". I just set the tone, and no one expects me to entertain them into a good mood, older children can be encouraged to find their happy heart or maybe they need more sleep.
It is hard to hear any child cry. We can use a timer, if its not to late to start early, use a timer, if a child is not "rescued" in 10 minutes they will give up and go to sleep.That is why monitors have off buttons, white noise is good for Mom and Dad too. I'm not talking hours, unless you have set really bad habits and they need to be broken.
Another thing my Mom taught me is that it's good to set a routine, go with babies basic schedule. If it is time for a nap, they should be in bed. Establish, tummy time, feeding times, play time area, and the designated bed area and time.
So new Moms, your baby needs a break, needs opportunity to grow, will not die if alone, will not suffer if they fuss for more that 5  minutes, and need to exercise their lungs with a bit of crying.
Always remember, God gave that baby to you, He did not intend it to be all stress, relax, enjoy!