If I understand correctly you have removed the old packages with "rpm -e apache/php/mysql.rpm" then you installed the new packages and they installed without error? yes?
So now you are attempting to get phpMyAdmin to work but you get an error and it says "Cannot connect to MySQL server". I am assuming that the rpm version doesn't set a default password so you have to set one yourself. To do this you enter this command:
mysqladmin -u root -h localhost password <password_Here>
After issuing that command in a shell, open the "config.inc.php" in the phpMyAdmin directory and enter the password around line 84 where it says:
$cfg['Servers'][$i]['password'] = '<password_Here>';
In theory this should all work. Also to clarify something the 'root' user on mysql is different then the 'root' user on your RH system, therefore also making the password different among other things.