I also wouldn't write a redirection, but I wouldn't rename the index.html either. The reason why I wouldn't is because if the default is to have a directory listing show up and PHP fails for whatever reason then it will show a directory listing instead of whatever the index.html file is.
Open your httpd.conf file and search for DirectoryIndex. This should register the load order of each file type if just a directory is requested (the index.php file for example). Alter this to move index.php to the left and index.html somewhere to the right of index.php. Restart your apache, and it should now serve your PHP files instead of html ones.
As of PHP 5.5, the MySQL library has been officially deprecated. It is recommended to move to either MySQLi or PDO libraries for your mysql connectivity. See here for help choosing which interface you prefer: http://php.net/manual/en/mysqlinfo.api.choosing.php