Passing value from select form with Ajax not working

You seem to be missing the ajax actions, there are two, one for logged in users and one for not logged in users.
Going by your action you will need to add the following actions.

// this one for logged in users
add_action('wp_ajax_my_action', 'my_action');

// this one for not logged in users
add_action('wp_ajax_nopriv_my_action', 'my_action');