how to save taxonomy terms in a custom user registration form in D7?
The column name for a taxonomy term reference is 'tid' rather than 'value', if you just change that your code should work:

$edit = array(
  'field_firstname' =&gt; array(LANGUAGE_NONE =&gt; array(array('value' =&gt; $form_state['values']['first']['field_firstname']))),
  'education' =&gt; array(LANGUAGE_NONE =&gt; array(array('tid' =&gt; $form_state['values']['second']['education']))),
);

user_save(drupal_anonymous_user(), $edit);