I had to change it up a bit to circle back to the Account ID.  Below is how I now have it but circling back is still a bit of an issue.  How do I get the list of Territories specific to the Account updated or created and add it to the Account custom field (text area)?

trigger AccountTerritories on Account (before update){
    if(Trigger.isBefore){
        Set<Id> setOfAccountIds = new Set<Id>();

        for(Account acct : Trigger.New){
            if(Trigger.isUpdate){
                if(acct.Id != null){
                    setOfAccountIds.add(acct.Id);
                }
            }
        }
 
        if(setOfAccountIds.size() > 0){

        /* Declaration of collection data types */
        Map<id,List<AccountShare>> MapOfAccountShare = new    Map<id,List<AccountShare>>();
       Map<Id, Id> mapOfGroup = new Map<Id, Id>();
       Map<Id, String> mapOfTerritories = new Map<Id, String>();


//Query in Account Share object
    /*
    Those Accounts which are assigned via Territory Assignment Rules.
You can query those Accounts by filtering RowCause = 'Territory' in AccountShare object query.
    */
    List<AccountShare> listOfAccountShare =
    [Select Id, UserOrGroupId, AccountId, RowCause from AccountShare where AccountId IN :setOfAccountIds];
    System.debug('=== contents of listOfAccountShare: ' + listOfAccountShare);
    System.debug('=== size of listOfAccountShare: ' + listOfAccountShare.size());
    List<Id> Groupids = new List<Id>();
    for(AccountShare a :listofAccountShare){
    Groupids.add(a.UserOrGroupId);
    }

    //Map of Account Share

  for(AccountShare acctShare : listOfAccountShare ){
        if (MapOfAccountShare.containsKey(acctShare.AccountID))
            {
        MapOfAccountShare.get(acctShare.AccountID). add(acctShare);
        }
        else{
        MapOfAccountShare.put(acctShare.AccountID, new  List <AccountShare> { acctShare });
        }  
        }  
    System.debug('=== all AccountShare keys in the map: ' + mapOfAccountShare.keySet());  
    System.debug('=== all AccountShare values in the map (as a List): ' + mapOfAccountShare.values());
    System.debug('=== size of mapOfAccountShare: ' + mapOfAccountShare.size());

    //Query in Group object            
    List<Group> listOfGroup = [Select Id, RelatedId from Group where Type='Territory' and Id IN :GroupIDs];
    System.debug('=== contents of listOfGroup: ' + listOfGroup);
    System.debug('=== size of listOfGroup: ' + listOfGroup.size());
//Map of Group object
    for(Group groupRecord : listOfGroup){
    mapOfGroup.put(groupRecord.Id, groupRecord.RelatedId);        
    }
    System.debug('=== all Group keys in the map: ' + mapOfGroup.keySet());  
    System.debug('=== all Group values in the map (as a List): ' + mapOfGroup.values());
    System.debug('=== size of MapOfGroup: ' + mapOfGroup.size());

    //Query in Territory object
    List<Territory> listOfTerritory = [Select Id, Name from Territory where Id IN :mapOfGroup.Values()];
    System.debug('=== contents of listOfTerritories: ' + listOfTerritory);
    System.debug('=== size of listOfTerritories: ' + listOfTerritory.size());
    //Map of Territory object
    
    for(Territory Territories : listOfTerritory){
    mapOfTerritories.put(Territories.Id, Territories.Name);
    System.debug('=== all Territories keys in the map: ' + mapOfTerritories.keySet());  
    System.debug('=== all Territories values in the map (as a List): ' + mapOfTerritories.values());  
    System.debug('=== size of MapOfTerritories: ' + mapOfTerritories.size());
    }
    for (Account updatedAccount : Trigger.new) {
    User territory = MapofTerritories.get(mapofGroup.get(mapofAccountShare.get(updatedaccount.id )));
    updatedAccount.territories = territory.name;
    }
 
}

}
}