How to prevent Tags showing at beginning of post, in Classic Theme?

Look for this this code in the themes posts template

<?php the_tags(); ?>

and remove it along with any div or p tags that are wrapped around it.

It is in the index.php on line 15 that contains:
<div class="meta"><?php _e("Filed under:"); ?> <?php the_category(',') ?> — <?php the_tags(__('Tags: '), ', ', ' — '); ?> <?php the_author() ?> @ <?php the_time() ?> <?php edit_post_link(__('Edit This')); ?></div>

This is what shows all the meta data about the post. Remove the whole line or individual items you do not want to show.

Or you could edit your style.css and add this line to it:

.meta {display:none;}