Adding WordPress header and footer to a PHP script

You mean this?

<?php get_header(); ?>

And for the footer:

<?php get_footer(); ?>

If you take a look at the get_header() function:

 function get_header( $name = null ) {
        /**
         * Fires before the header template file is loaded.
         *
         * The hook allows a specific header template file to be used in place of the
         * default header template file. If your file is called header-new.php,
         * you would specify the filename in the hook as get_header( 'new' ).
         *
         * @since 2.1.0
         * @since 2.8.0 $name parameter added.
         *
         * @param string $name Name of the specific header file to use.
         */
        do_action( 'get_header', $name );

        $templates = array();
        $name = (string) $name;
        if ( '' !== $name ) {
                $templates[] = "header-{$name}.php";
        }

        $templates[] = 'header.php';

        locate_template( $templates, true );
}

Do you have at least a header.php file on your root theme?

A common Wordpress structure:

your_project_folder
    -wp-admin
    -wp-content
        -languages
        -plugins
        -themes
            -YOUR_THEME_FOLDER
                -[HERE YOU PLACE YOUR index.php file and header.php for example, and inside index.php you place your get_header() function]
        -upgrade
        -uploads
    -wp-includes
    index.php
    wp-activate.php
    wp-blog-header.php
    wp-comments-post.php
    wp-config.php
    wp-cron.php
    [...more files]