Just download twentyten and see how it works, it’s relatively simple to use it as an example. Nothing needs to go into functions.php for comments to work. You can use custom callbacks in there if you wish, but just a few template tags get the comments going
<?php comments_template( '', true ); ?>
Is what goes into your single.php, page.php etc to disply the form.
http://codex.wordpress.org/Function_Reference/comments_template
Something like
<p><?php comments_popup_link('No comments yet', '1 comment so far',
'% comments so far (is that a lot?)', 'comments-link', 'Comments are
off for this post'); ?></p>
Is great for index.php
http://codex.wordpress.org/Function_Reference/comments_popup_link
And then within comments.php, wp_list_comments displays the comments, and yes,
<?php comment_form(); ?>
Is what displays the actual form
I like this article for usage of the comment form