Trigger to update parent object value with child value

Just looking at the code you have there a few things stand out. I haven't tested this but I've added comments which might help you along

trigger UpdateAmount on Quote__c (after insert, after update) { //You want it on update too, right?
  Map<ID, Opportunity> parentOpps = new Map<ID, Opportunity>(); //Making it a map instead of list for easier lookup
  List<Id> listIds = new List<Id>();

  for (Quote__c childObj : {

  //Populate the map. Also make sure you select the field you want to update, amount
  //The child relationship is more likely called Quotes__r (not Quote__r) but check
  //You only need to select the child quotes if you are going to do something for example checking whether the quote in the trigger is the latest
  parentOpps = new Map<Id, Opportunity>([SELECT id, amount Name,(SELECT ID, Total_List_Price__c FROM Quotes__r) FROM Opportunity WHERE ID IN :listIds]);

  for (Quote__c quote: Trigger:new){
     Opportunity myParentOpp = parentOpps.get(quote.Opportunity__c);
     myParentOpp.Amount = quote.Total_List_Price__c;

  update parentOpps.values();

Late to the party, but...

I wouldn't use a trigger unless absolutely necessary. This is a great use-case for Lightning Process Builder. It can update the parent record with a value from the child record. And without code!