trigger updateCM on Account_Membership__c (after update) // if you make AFTER, you cannot fired old values
{
  // first, we need to change the status of all case roles if the
  // CM status value has changes
  
  if (Trigger.isUpdate) // no need to do this when trigger fires update only
  {
    for (Integer i = 0; i < Trigger.new.size();i++)
    
    {
      Account_Membership__c casenew = Trigger.new[i];
      Account_Membership__c caseold = Trigger.old[i];
      
      // we check to see if the status has changed
      if (casenew.Status__c != caseold.Status__c) 
      {
            for(Contact_Membership__c cr: [select Id, Account_Membership__c  from Contact_Membership__c where Account_Membership__c = :casenew.id ])
            // try to not use query into the for loop
            {
                cr.Status__c = casenew.Status__c;
                cr.Type__c =  casenew.Type__c;
                cr.Joined__c  = casenew.Joined__c;
                cr.Expires__c = casenew.Expires__c;
                cr.OwnerId =  casenew.OwnerId;
                update cr; // try to not use dml call into the for loop
            } 
           
      }          
      }
    } 
  }

trigger updateCM on Account_Membership__c (before update) 
{
    Map<Id, Account_Membership__c> amForQuery = new Map<Id, Account_Membership__c>();
    List<Contact_Membership__c> cmToUpdate = new List<Contact_Membership__c>();

    // get Account_Membership for who Status__c changed  
    for (Account_Membership__c amNew : Trigger.new)
    {
        if (amNew.Status__c != System.Trigger.oldMap.get(amNew.Id).Status__c)
        {
            amForQuery.put(amNew.Id, amNew);
        }
    }    

    // query Contact_Membership
    for(Contact_Membership__c cr : [SELECT Id, Account_Membership__c  
                                    FROM Contact_Membership__c 
                                    WHERE Account_Membership__c IN :amForQuery.KeySet() ])
    {
        cr.Status__c = amForQuery.get(cr.Account_Membership__c).Status__c;
        cr.Type__c =  amForQuery.get(cr.Account_Membership__c).Type__c;
        cr.Joined__c  = amForQuery.get(cr.Account_Membership__c).Joined__c;
        cr.Expires__c = amForQuery.get(cr.Account_Membership__c).Expires__c;
        cr.OwnerId =  amForQuery.get(cr.Account_Membership__c).OwnerId;
        cmToUpdate.add(cr);   
    }
    
    // DML statement
    if(cmToUpdate.size() > 0)
        update cmToUpdate;

}