Modify Homepage Search (add a category)

I’m trying to modify the search fields on homepage (using Sep 2016 child theme).

I realize it’s the home.php file I need to edit, I need to know how to add new categories as options to select.

This is the website: https://programs.studying-in-germany.org/

I’m trying to get the form fields to look like this sidebar form: https://programs.studying-in-germany.org/?s

I’d appreciate any help.

Thanks!