Is it true that after a divorce the mother is not allowed to take the children out of the state without the father's consent?

Michigan statutes speak to the issue on different levels. Generally speaking, a parent can take the child out of state for vacations or trips without court permission and only advising the other parent as a courtesy.  But a parent can not change the child's permanent residence out of state unless the court has provided prior written permission to that parent. Seek legal help.  Good luck.

It depends on what your Order states.  If there is a geographical restriction, then the child has to live within the geographic area listed in the Order.  This kind of restriction can only be lifted by agreement, or by Court Order.  If you are just talking about a visit or vacation, as long as it doesn't interfere with Dad's visitation, he doesn't have the right to object.  It will probably help everyone's relationship if you at least notify Dad of the time out of state.  You cannot unilaterally change Dad's visitation time because you have a vacation planned.