hi there -
I actually had a baby in Spain, and the health care is great for babies. The best and easiest way about it is to use the National Health Care. You just have to get a card for your baby - go to the Centro de Salud in your neighborhood and they can tell you where to go to the the card. Once you have the card, you an go to your Centro de Salud and ask for an appt. with a pediatrician. From then on, he/she will be your standard pediatrician plus you will also get a nurse assigned to you (for vaccinations). All medical care is covered and free. It's all very easy and we were really happy with our pediatrician. (Unfortunately we are now living in Greece where things are very different and not nearly as good).
Hope that helps -
Lindsay