Twig template for search results page
but I'd like to actually modify the buttons and text fields at the top
  of the search page.


They do not have templates by default, because they are forms. You could create a twig template for the search form by follow this tutorial: How to manipulate forms in Drupal 8

But in my opinion, it is easier to use hook_form_alter on your .theme file and add the classes here.

This is how, the drupal default theme, Bartik changes the search button to a magnifying glass icon by adding a custom class to the submit button in bartik.theme

/**
 * Implements hook_form_alter() to add classes to the search form.
 */
function bartik_form_alter(&amp;$form, FormStateInterface $form_state, $form_id) {
  if (in_array($form_id, ['search_block_form', 'search_form'])) {
    $key = ($form_id == 'search_block_form') ? 'actions' : 'basic';
    if (!isset($form[$key]['submit']['#attributes'])) {
      $form[$key]['submit']['#attributes'] = new Attribute();
    }
    $form[$key]['submit']['#attributes']-&gt;addClass('search-form__submit');
  }
}




You can use Devel and use dpm($form) to see what's inside the $form array. 

Example: 

/**
 * Implements hook_form_alter() to add classes to the search form.
 */
function bartik_form_alter(&amp;$form, FormStateInterface $form_state, $form_id) {
  if (in_array($form_id, ['search_block_form', 'search_form'])) {
    $key = ($form_id == 'search_block_form') ? 'actions' : 'basic';
        drupal_set_message(dpm($form));
  }
}


Make sure you don't have the search block on the search page, other wise it will print the array for both forms and might confuse you which one is which.