How to update the page breadcrumb when using theme page
If you are looking for GUI to handle breadcrumbs, take a look at Custom breadcrumbs module.
Otherwise, you can manage your breadcrumbs programatically in lot of hooks. The code you've printed is correct, so you can create for example a function with a switch case like this one :

function manager_breadcrumbs($form_id) {
  $breadcrumb = array();
  $breadcrumb[] = l('Home', '&lt;front&gt;');
  switch ($form_id) {
    case 'my_first_form' :
      $breadcrumb[] = l('Category', 'taxonomy/term/1');
      break;
  }
  drupal_set_breadcrumb($breadcrumb);
}


And then, call this function in a hook_form_alter() like this.

function MY_CUSTOM_MODULE_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  manager_breadcrumbs($form_id);
}