Remove alphabetical order from wordpress categories

Here you go: just place this in your functions.php:

/**
 * Sort post_tags by term_order
 *
 * @param array $terms array of objects to be replaced with sorted list
 * @param integer $id post id
 * @param string $taxonomy only 'post_tag' is changed.
 * @return array of objects
 */
function plugin_get_the_ordered_terms ( $terms, $id, $taxonomy ) {
    if ( 'post_tag' != $taxonomy ) // only ordering tags for now but could add other taxonomies here.
        return $terms;

    $terms = wp_cache_get($id, "{$taxonomy}_relationships_sorted");
    if ( false === $terms ) {
        $terms = wp_get_object_terms( $id, $taxonomy, array( 'orderby' => 'term_order' ) );
        wp_cache_add($id, $terms, $taxonomy . '_relationships_sorted');
    }

    return $terms;
}

add_filter( 'get_the_terms', 'plugin_get_the_ordered_terms' , 10, 4 );

/**
 * Adds sorting by term_order to post_tag by doing a partial register replacing
 * the default
 */
function plugin_register_sorted_post_tag () {
    register_taxonomy( 'post_tag', 'post', array( 'sort' => true, 'args' => array( 'orderby' => 'term_order' ) ) );
}

add_action( 'init', 'plugin_register_sorted_post_tag' );

(Credit goes to lgedeon on the WordPress Core Trac)

Now all you need to do is enter the tags in the order you want them to appear.

Note: the above code shows how to do this for post_tag. If you want a different taxonomy, just update the above code with the taxonomy name you want.