Yes, pediatrician is specialist doc for children/baby.
All babies have an assigned doc at hospital after delivery, and for followups. But you can choose your own fav pediatrician or one near your place for convenience sake. In general, you bring baby there for jaundice (during 1st month) and/or health assessment cum vaccinations.
Consultation price has increased fairly over this couple yrs.. Neighborhood to popular clinic can range from $40+ to $60+ , or more for famous ones.
Hope this helps