Looking for a way to take readers to random post when clicking a link

Looks like you’re after Random Post Link.

For completeness here’s some code (the plug-in is a little more involved as it stores as a cookie previous random posts, so they don’t twice).

(The following is based on the above plug-in by Scribu)

//Create random url - use this where you want the link to be displayed
$url = add_query_arg( 'wpse82608', 'random', trailingslashit( get_bloginfo( 'url' ) ) );
printf( '<a href="https://wordpress.stackexchange.com/questions/82608/%s" > Random Post </a>', $url );

Then interpet the random url (this can go in a plug-in or, if you must, your theme’s functions.php)

add_action('init', 'wpse82608_random_post_redirect' );
function wpse82608_random_post_redirect(){

    if( empty( $_GET['wpse82608'] ) || 'random' != $_GET['wpse82608'] )
        return;

    //Get a random post
    $posts = get_posts( array(
        'orderby' => 'rand',
        'showposts' => 1,
     ));

    //If no posts founds - redirect to site.
    if ( empty($posts) ){
        wp_redirect( get_bloginfo( 'url' ) );
        exit();
    }

    //Get the random post's ID
    $id = $posts[0]->ID;

    //Redirect to post
    wp_redirect( get_permalink( $id ) );
    exit();
}

This is untested.