Simple jQuery UI Dialog with WordPress!?

Hi together,

I think I am a bit lost in the depths of registering, enqueing and executing JS in WP: All I want to do, is to display on a page like this.

I understand, that all the required JS (jQuery UI) comes with each standard WP installation, correct?

I found the following code snippets in the Codex documentation:

<?php
function my_init_method() {
    if (!is_admin()) {
        wp_enqueue_script( 'jquery' );
    }
}    

add_action('init', 'my_init_method');
?>

Do I need to register the jQuery / jQuery UI script at all – or is this regsitered by default?

How do I enqueue the script on the pages I require it? Just like above? Do I need to make sure to check, if it’s not already enqueued by another script?

And last but not least, how do I include my intended behaviour then? I understand, I need a snippet like the one below, but how do I include that on the page?

jQuery(document).ready(function(){
    jQuery(#somefunction) ...
});

Maybe there is a tutorial somewhere explaining that comprehensively, then I would really appreciate to point me there!

Thanks!

Best regards
th23