Not taking full advantage of all the tax/employer benefits: Most people only think of insurance and 401(k)s when they think about their employer benefits. A lot of employers offer much more than that — a flexible spending account, dependent care account, gym memberships, discounts to stores and even subsidized child care. Find out all the benefits you are eligible for and take advantage of them.
Failure to plan: When the baby comes, it is pretty much “stop everything else that is going on in your life and take care of the baby.” It is a lot easier and better to make a plan as soon as you know you are going to have a baby. The number of options available to save money might also go down the longer you wait.
Those are the big money mistakes of which I am aware. There are also other things like missing the deadline for a bill which can be fixed by planning ahead and automating bill pay, but from personal experience most of the companies will understand and waive the late fee once if you explain your new life change.