Modify the searchform.php, or create one in a child theme (recommended), and you should be able to modify the form all you want.
The default HTML is in /wp-includes/general-template.php, to which I have already linked, but don’t edit that. It is a Core file. Copy the markup into your searchform.php if you need a place to start.
Please refer this link to modify search function detail.
Also refer this link for adding custom post type in search.