How to edit theme functions file to modify pagination?

Just specify an image as the “nextpagelink” and “previouspagelink” instead of the << or >>:

<?php wp_link_pages(array('before' => '<div class="pagenav"><strong>Navigate</strong>', 'after' => '</div>', 'next_or_number' => 'number', 'nextpagelink' => __('<img src="https://wordpress.stackexchange.com/questions/1314/PUT YOUR IMAGE URL HERE" />'), 'previouspagelink' => __('<img src="https://wordpress.stackexchange.com/questions/1314/PUT YOUR IMAGE URL HERE" />'))); ?>

Also, you are correct that by default you are limited to either “Numbers” or “Next/Previous” links but a plugin can extend this: http://wordpress.org/extend/plugins/wp-pagenavi/

Bonus: Add this to your function.php and it will add a “Nextpage” button next to the “More” button in the WYSIWYG editor:

//  Add Next Page Button to TinyMCE Editor
add_filter('mce_buttons','wysiwyg_editor');
function wysiwyg_editor($mce_buttons) {
    $pos = array_search('wp_more',$mce_buttons,true);
    if ($pos !== false) {
        $tmp_buttons = array_slice($mce_buttons, 0, $pos+1);
        $tmp_buttons[] = 'wp_page';
        $mce_buttons = array_merge($tmp_buttons, array_slice($mce_buttons, $pos+1));
    }
    return $mce_buttons;
}