Well I figured out the problem by comparing the syntax of these php files to the php files for another plugin. I made the following changes:
1) Change each
2) Save the return of the mysql_connect and pass it as the last argument to mysql_select_db, mysql_query, and mysql_close.
So for the index.php, the first part becomes:
@mysql_select_db($database, $sql) or die( "Unable to select database");
$query="SELECT `steamId`, `nickNames`, `nFinnished`, `rankTotal`, `primaryRank` FROM `skillrank` ORDER BY `nickNames` ASC";
Also I moved the dbinfo.inc.php to a subdirectory (had to update the include statements in the php) so I can limit its access.
This is very important because it has your database root password!
Works like a champ now!