Can my child's father prohibit me from moving out-of-state?

If there is no court order in effect, than you you can take your child out-of-state.  However, your child's father can go to court in your present state and obtain an order which will require you to return there.  If you don't return you can be charged with parental kidnapping.

The safest way for you to obtain legal custody of the children is for you to go to court first.  What you want to do is get an order from the court on custody/visitation.  The court will consider the bests interests of the children in making its determination.  It will look at such things as the age of your child, the relationship of the parents and the child, the child's present living situation, the reason for the move, etc. 

Note:  You could file for custody once you re-establish residency in your new state (but that will take time).  In the meantime, your child's father could file and, as stated above, you then could be required at that point to return.