In most states, there are no charges incurred for changing the last name of a child during the course of a Family Court proceeding. However, a $65 fee is charged for filling for a name change in Civil Court and a $210 fee for Supreme Court. If the father has a joint custody, you need to have his consent. Check out http://www.liftonline.org/guides/pdf/guide_102.pdf.
Probably depends on where you live. I changed my name last year and it was about $80, then the cost of a new licence, and passport. And as far as I know you do need consent from the father if you have joint custody