How to set cookies to switch style sheets when a link is clicked and set it till another particular link is clicked?

You can use init action to set cookie, and other init action to read it. However due its a cookie… you will need also to check GET variable.

add_action('init', 'is_it_mobile_or_desktop', 1);

function is_it_mobile_or_desktop(){
    if (isset($_GET['site']) && in_array($_GET['site'], array('mobile', 'desktop')) ) {
        setcookie( 'site', $_GET['site'], time() - YEAR_IN_SECONDS, SITECOOKIEPATH, COOKIE_DOMAIN );
    }
}

add_action('init', 'who_am_i', 2);
function who_am_i(){

    $site = isset($_GET['site']) && in_array($_GET['site'], array('mobile', 'desktop')) ?  $_GET['site']
            : (isset($_COOKIE['site']) && in_array($_COOKIE['site'], array('mobile', 'desktop'))  ? $_COOKIE['site'] : 'default_case');

    if ($site == 'mobile'){
        function responsive_css(){
            wp_enqueue_style(
                'wpa_custom',
                get_template_directory_uri() . '/css/responsive.css'
            );
        }
        add_action( 'wp_enqueue_scripts', 'responsive_css', 999 );
    } else {
        function default_css(){
            wp_enqueue_style(
                'default_css',
                get_template_directory_uri() . '/style.css'
            );
        }
        add_action( 'wp_enqueue_scripts', 'default_css', 999 );
    }
}