I wish to add the publish tags into its div class in order to add specific CSS for various kinds of tags.

If you wish to output the tag title(s) of this article being seen right into a div, do this :

<div class="
<?php
$posttags = get_the_tags();
if ($posttags) {
  foreach($posttags as $tag) {
    echo $tag->name . ' '; 
  }
}
?>
">
...
</div>

If you wish to output the tag title that's being display (let us the tag's page), do this :

<div class="<? echo single_tag_title(); ?>">
.....
</div>

Saying yes with mrtsherman, this is vague, but when your attempting to title a category you'd function something similar to

<div class="the-name-you-want">
</div>