Convince them to use Debian so you can use apt.
...
More seriously:
1. Packages
On the DB server, you need mysql-server. That's it. You probably want the client there as well, for setup and trouble shooting. I'd seriously consider putting on Apache and PHP to run PHP MyAdmin as well, though you don't have to do anything but mysql-server.
On the webserver, you want Apache, PHP, PHP-MySQL, MySQL client. Depending on the nature of the use of this webserver (internet vs intranet, timeliness of data, etc), you may want to seriously consider squid to cache your generated HTML/images.
I don't know the package/RPM names because my servers are all running Debian and Ubuntu rather than Red Hat or other RPM-based distro (entirely due to package management differences).
2. Setup
Open an SSH session or two to each server. On the DB server, use yum to install the mysql-server. On the webserver, use yum to install the other stuff. In the spare SSH sessions, start looking (Looking!) at how the network is configured. Make sure they don't use DHCP, or if they do, that there's a DNS server kept up-to-date with the host names. Better to use static addresses and hardcode the other machine's name in /etc/hosts (just remember you've done so!). If you need to make changes, wait till yum is finished.
I hope the network is setup properly, remotely configuring a network can be hairy if you don't do it right. SSH will auto-restore if you use 'ifdown -a; ifup -a;' to reload the ethernet. Anyway, you should probably sort this out now.
Just proceed as normal for the LAMP stack on the webserver, except in your PHP app, the DB connect calls need a host parameter, rather than the default of localhost. That's it.
On the DB server, you may need to configure MySQL to accept connections from remote hosts - once that's done, connect to the DB. If you installed the client on this machine, connect as normal; if on the webserver, just add a '-h dbserver.local' to the command to talk to the db-server.
Create a user for your PHP app as normal, but put the webserver's address in the command:
GRANT INSERT,SELECT,DELETE,UPDATE,... to 'foo'@'webserver.local' IDENTIFIED BY ...
Test.
3. Upgrades
I don't remember how to upgrade Red Hat from the command line, sorry. Presumably there's a yum command though.