Drupal - How do I alter the submission button class?

If I were you, I would not change the class name but rather add your own class to the form element. You can do this by implementing hook_form_alter in a custom module or (in case of Drupal 7) in your theme. The code would look something like this:

<?php
function mymodule_form_alter(&$form, &$form_state, $form_id) {
  if ($form_id = 'my_form') {
    $form['actions']['submit']['#attributes']['class'][] = 'form-submitone';
  }
}
?>

By the way, the submit button element is not necessarily $form['actions']['submit']. You will have to figure out which element is the submit button in your specific situation. You can do that by outputting the $form variable; I recommend installing the Devel module and adding dsm($form); to the function above.