Let me clear some things up for you
Residency: What you do immediately after medical school. This is what you specialize in. There are residencies in pediatrics.
Fellowship: What you do after residency. This is what you subspecialize in.
There are fellowships that you can do after a pediatrics residency to become a neonatologist, a pediatric cardiologist, etc. Most are not that competitive because with a few exceptions, pediatric subspecialists don't make that much more than pediatricians who don't subspecialize.
There are also pediatric fellowships you can do after completing a residency in another field of medicine. For example, you could do a two-year pediatric surgery fellowship after completing a general surgery residency. These also vary. Pediatric general surgery is considered an extremely competitive fellowship. Other specialties' pediatric fellowships may not be.