A parent generally needs permission from the other parent before travelling with the child especially if the travel will impact the parenting time of the other parent. However, if the separation agreement dictates how travel should occur, both parents are expected to follow the terms of the agreement.
You need to obtain a passport for your child if he/she will be traveling overseas. Your country has requirements for getting a birth certificate for your child, and your government's passport office will tell you what other documents may be used as alternatives.
What is included in a Child Travel Consent Form? the parents' and/or legal guardians' name(s) the child's or children's name(s) details of the adult (and, where applicable, organisation) accompanying the child. the child's medical information. the parents and/or legal guardians' emergency contact details.
In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.