I want to get re-married as soon as I get the divorce judgment. Can I relocate with my child without my ex's permission?

Relocation is a tricky issue and there doesn’t seem to be a set rule. However, most New York divorce and family law attorneys will talk about the Tropea case which is the seminal case on the issue of relocation. However, In Tropea the the ex-wife was pregnant and engaged, and requested permission to move the children from Onondaga county to Schenectady New York because her new husband had a good job there. The court allowed her to move.
But there are other cases where, even though there was a re-marriage and an economic advantage to moving, the court did not allow the move. Ultimately the court is going to consider what is the best interest of the child. The court will look at who is the main care-giver of the child. The court will consider the non-custodial parent’s relationship with the child, and whether, post divorce, said parent was excercising his or her visitation rights. The court will consider whether the new relationship really needs to move out of the proximity of the former spouse, or whether an equally well paying or comparably well paying job can be found right here in New York.
Yes, sometimes, the court will allow the custodial parent to move to another country with the children. And sometimes, the court will not allow the custodial parent to move to a nearby state. It really depends on a lot of different factors.
I hope this helped.