In an ideal world, you would start saving for the baby before you are pregnant. Once you have negotiated your maternity leave and have an idea of your income, create a new budget that includes baby expenses.
Also check whether you qualify for any tax credits such as the Child Tax Credit, the Child and Dependent Care Credit, or the Adoption Credit and Adoption Assistance Programs. Taking out a College 529 savings plan for your child will reduce your taxable income.