Under the advanced tab you can see what templates you need to override the views template. Copy the templates mentioned from the modules/views/themes folder to you own theme/templates folder, rename them as the view suggests under advanced and style your output.
note: if you display a teaser or full node, but forced it to use fields, the styling (making it h3) won't work.
The h3 is coded in views-view-unformatted-list.tpl.php. The $title there is output of the field you grouped by. There you can set it to h2.
UPDATE: This is the default views template for outputting lists (the container which contains all rows as a group). This one is named views-view-unformatted.tpl.php, you need to rename it to match with your view.
<?php
/**
* @file
* Default simple view template to display a list of rows.
*
* @ingroup views_templates
*/
?>
<?php if (!empty($title)): ?>
<h3><?php print $title; ?></h3> // THIS IS THE GROUPNAME
<?php endif; ?>
<?php foreach ($rows as $id => $row): ?> // THESE ARE THE ROWS INSIDE THE GROUP
<div<?php if ($classes_array[$id]) { print ' class="' . $classes_array[$id] .'"'; } ?>>
<?php print $row; ?>
</div>
<?php endforeach; ?>
If you want to put style on ALL the rows inside the group then you can use the view-ui itself, or use the template called views-view-fields.tpl.php to override all rows.
If you want to style ONE specific row use views-view-field.tpl.php (note there is no 's' behind field).