Registering with your health insurer
Whatever you do, do not forget to register your baby with your health insurer. Preferably do it in the first few weeks; it must be done within 4 months of the birth in any case. The insurance then applies retroactively from the day of birth.
Children under 18 are insured free of charge and pay no excess. You add your child to the policy of either parent. That parent's supplementary insurance then covers your child too, so consider which policy suits best. The insurer will ask for your child's citizen service number (BSN), which you receive when registering the birth, so that comes first.
What else to arrange
After you register the birth, the Sociale Verzekeringsbank (SVB) automatically sends you a letter about child benefit; there is nothing to do until that letter arrives. Also check on toeslagen.nl (in Dutch) whether your benefits change, such as the child budget or childcare benefit. And you will receive an invitation from the child health clinic. The full checklist is on rijksoverheid.nl (in Dutch).