First off, when you type "mysql" at the prompt that's simply the client you're starting. You need to have the server running as well (when the server starts it creates the sock file). It has been years since I used an rpm based distribution, but I think that the command
rpm -qa | grep mysql will list all the mysql-related packages installed. You should probably be looking for "mysql-server-...".
RedHat has its own tools for starting and stopping services and defining what services to launch at boot and so on. If you search the RedHat/Fedora section of this site I am sure you will find hints on starting and stopping it.
Oh, and if you really are running RedHat 8 you should upgrade as soon as possible. Running an old non-supported distribution is a sure way of getting into security related problems quickly. A distribution very popular among Linux newbies and veterans alike is
Ubuntu, a Debian derivative, you might want to check that out if you're not comfortable with Fedora Core.
Håkan