There's one small issue: your code doesn't support bulk processing, which means that a synchronization against Salesforce might fail. Your SOQL query should be moved outside of the loop.

trigger Update_FirstCompletedActivity_onOpp on Task (after insert, after update) {
  // Map tasks to the related opportunity.
  Map<Id,Task>oppTaskMap = new Map<Id,Task>();
  // Opportunities to update.
  List<Opportunity> opps = new List<Opportunity>();
  // Build the mapping for closed tasks.
  for(Task t:Trigger.new)
   if(t.whatid<>null && t.Status=='Completed')
     oppTaskMap.put(t.whatid,t);
  // Find opportunities that will be updated, and add them to the list.
  for(Opportunity o:[select id,first_completed_activity__c from opportunity where id in :oppTaskMap.keySet() and first_completed_activity__c=null])
    opps.add(new opportunity(id=o.id,first_completed_activity__c=oppTaskMap.get(o.id).DateTimeCompleted__c));
  // If we have any updates, do so now.
  if(!opps.isEmpty())
    update opps;
}

In regards to code coverage, you should probably take the path of least resistance. I would write a class as follows:

@isTest
public class testTriggers {
  public static testMethod void testTaskTriggers()
  {
    Opportunity o = new Opportunity(Name='Test',CloseDate=System.Today(),StageName='Won');
    insert o;
    Task t1 = new Task(Status='Not Started',Subject='Test',WhatId=o.id,Type='Test',ActivityDate=System.Today().addDays(10));
    insert t1;
    o = [select id,First_Completed_Activity__c from opportunity where id = :o.id];
    System.AssertEquals(null,o.First_Completed_Activity__c,'This date should be a null value.');
    t1 = [select id,whatid,activitydate,status,datetimecompleted__c from task where id = :t1.id];
    t1.Status='Completed';
    update t1;
    System.AssertEquals(t1.DateTimeCompleted__c,o.First_Completed_Activity__c,'This date should be set to the Task date.');
    Task t2 = new Task(Status='Completed',Subject='Test',WhatId=o.id,Type='Test',ActivityDate=System.Today().addDays(5));
    insert t2;
    o = [select id,First_Completed_Activity__c from opportunity where id = :o.id];
    System.assertEquals(t1.ActivityDate,o.First_Completed_Activity__c,'This date should be set to the first Task, not the second.');
  }
}

We test the system by confirming that the date field is null at first, then set to the first task's date, and then remains so for the second insert. This is generally how a test method should go. Your trigger should have 100% coverage with this simple class method. The idea of testing is to make sure that each part of the code is covered.

You can use Run All Tests in Sandbox, then click on the Percent Complete column to see which lines of code were missed. Please let us know if you need further help with the code.