If you want to change the look of the Joomla search module and Joomla search component:
Copy the file:
/components/com_search/views/search/tmpl/default.php
to
/templates/my_template/html/com_search/search/default.php
Now make a few changes in that file and refresh your site.
In a similar fashion, you can also override the Search Module layout file. Copy the file:
/modules/mod_search/tmpl/default.php
to
/templates/my_template/html/mod_search/default.php