How can I show some standard html code across any theme I install?

You want to create a new plugin for this. See: http://codex.wordpress.org/Plugin_API/Action_Reference/wp_head

1. Create a new file in /wp-content/plugins/ called headerstuff.php (or whatever)

2. Drop the following code in it:

    <?php
    function header_code() {
    $output .= ""; //code segment 
    echo $output;
    }
    add_action('wp_head', 'header_code');
    ?>
   

3. Add your code between the quotation marks on the line starting with “$output”.

4. Activate, done!