Are you giving birth in the private or public system? For public, they will have a pediatrician do the needful and arrange follow-up visits for consultaton/vaccinations. You can choose to go private for the vaccinations/follow-up after giving birth in public - that's what we did. Then, you just call up a pediatrician of your choice and make an appointment.
For private, each hospital might be different. At Baptist, my understanding is that they will have a pediatrician on call who will attend to the baby after delivery.
Whether insurance covers it or not depends on the plan. I'm told my husband's staff insurance will NOT cover maternity but anything subsequent to delivery relating to the baby will be covered. So even for my first baby, all his vaccinations at the private pediatrician were covered even though he didn't even have an insurance card yet. For this one, I'm looking into separating the bills relating to maternity and anything relating to treatment of the baby for insurance purposes.