How register library to use jquery correct

Question

You should use the built-in version of jQuery for compatibility reasons.

All you need to do is to enqueue it:

function theme_scripts() {
    wp_enqueue_script('jquery');
}
add_action('wp_enqueue_scripts', 'theme_scripts');

Or just add it as a dependency for your script:

function theme_scripts() {
    wp_enqueue_script( 'my-script', <PATH>, array('jquery'), ... );
}
add_action('wp_enqueue_scripts', 'theme_scripts');

But then… you have to remember, that jQuery is run in no-conflict mode, so in your scripts you have to access it by jQuery and not by $. So your script should look like so:

jQuery(function ($) {
    // here you can use $ sign to get jQuery, because of the param of function

    $('a').click(...); // so this will work OK
});

// but here you can't and you have to use jQuery
jQuery('a').click(...); // this will work also

$('a').click(...); // but this won't, because there is no $ accessible here