There most likely is a mysql script in /etc/init.d but you don't have it selected to start at boot up. Does the command 'service mysql start' or 'service mysqld start' bring up the service. You could use the 'chkconfig' command as root to start the mysql daemon to start when you boot up.
chkconfig --list will print out a table of services and the runlevels they start in. It also gives the instructions to use to start a service on a certain run level. You probably want to start 'mysqld' under runlevels 3 and 5. There may be a second related service, but I'm not at my linux box to check. 'info mysql' may give more information.
Here is a skeleton InitV script for redhat:
If you notice the comment line in the script:# chkconfig: 2345 55 25
This line is used by chkconfig, so your script needs it. In other words, this is more than a comment line.
The scripts in /etc/init.d/ are linked to /etc/rc.d/rc.3 for example.
The number after 'rc.' is the run level where these scripts will start.
Scripts beginning with the letter 'S' are started during boot up in alphabetical order. Scripts starting with the letter 'K' are run during shutdown. Again, these are links to the scripts in /etc/init.d (same as /etc/rc.d/init.d, i.e. one directory location is a symbolic link to the other. )
There is also an /etc/rc.d/rc.local script that will run after the rc.d scripts run. However for the mysql server, you probably want to use the script that the system runs in /etc/init.d/ .
There may be a red hat gui'd equivalent to the 'chkconfig' program, for starting services.