Once your baby is born the umbilical cord will still be attached to the placenta and the doctor will put a clamp to it. He may cut it or let Dad cut the cord, if he is not too squeamish.The clamp is kept for 24-48 hours till it has dried and upon discharge they take it off. The umbilical stump usually falls off with 1-3 weeks. They will then bring the baby to the warmer to dry and stimulate your baby till he/she cries. Vital signs will be checked to make sure baby is stable. An eye ointment will be applied to the baby to prevent infections that he/she might be exposed in transit through the birth canal. Since babies have low vitamin K level which results in clotting. Your baby will be given a shot of Vitamin K to prevent baby from bleeding. If everything is good and mother is awake they give the baby to mother so she can try to nurse.
Another more important thing they will do is to put some matching bracelets on the baby and the mom. Making sure that it is the right baby for the right mom.They also would get some baby footprints for documentation. They will then call the pediatrician who will then check the baby within 24 hours of birth. They then would check the baby everyday that the mom will still be in the hospital. An alarm will be applied either to the baby's umbilical clamp or the ankles. This is what you call a "baby low-jack". if somebody attempts to get your baby without permission the elevators would not close or the alarms on the doors will be set off. This is a great way of protecting your babies from abductions.
You have the nurses in the hospital for support. Ask a lot of questions because they are very knowledgeable. Get their help as much as you can then you can go home more confident.