After installing Drupal, there are files in the root directory that I should delete. I know that install.php is one of those. Which other files should I delete?
You shouldn't really delete any files. If you really wanted to, you could delete the install file and various txt files, but that's about it. A better solution if you are afraid of security is to not let the files be accessed through the web server. Drupal only use the index.php file for serving content.
The upgrade advice is the same as for installations.
The last step in an upgrade is to delete or move the following files from your site:
install.php CHANGELOG.txt INSTALL.txt INSTALL.mysql.txt INSTALL.pgsql.txt LICENSE.txt MAINTAINERS.txt UPGRADE.txt
You just need to Change the file permissions if you wish for files like config.php but you don't need to delete any file, like some premium forum installations.
You shouldn't remove any files. You cannot obscure the fact that your site is running on Drupal. See: https://www.drupal.org/node/766404