Drupal 8 route not found
In first_module.routing.yml, controller there is no src and you use 2:  ::

first_module_hello_world:
    path: '/hello-world'
    defaults: 
        _controller: '\Drupal\first_module\Controller\TrainingModule::helloWorld'
        _title: 'hello world'
    requirements:
        _permission: 'access content'


Also without src in drupal 8 there si no need to use  return $output;
Alsi class is not good. Class is TrainingModule not TrainingModuleControlle and file need to cole the some TrainingModule

namespace Drupal\first_module\Controller;
use Drupal\Core\Controller\ControllerBase;

/**
 * Controller routines for hello_world module routes.
 */
class TrainingModule extends ControllerBase {

 /**
  * Return the 'Hello World' page.
  *
  * @return string
  *   A render array containing our 'Hello World' page content.
  */
 public function helloWorld() {

  return array(
    '#type' =&gt; 'markup',
        '#markup' =&gt; $this-&gt;t('Hello World!'),
  );
 }
}


change first_module_hello_world: to first_module_hello_world.help:

make the file first_module_hello_world.module and write this code

function first_module_hello_world_help($route_name, $route_match) {
 switch ($route_name) {
  case 'first_module_hello_world.help':
    $output = '';
    return $output;
 }
}