Usually a MySQL installation will also create some default databases. In case they don’t exist you can create the default databases by running this:
mysql_install_db
Setting a MySQL root password
The most important thing is to ensure that all root accounts have set passwords. For instance this is the default mysql database after a normal MySQL installation.
This a very insecure setup.
To learn how to setup a password for the MySQL root accounts or to delete the root users entirely see this short referent on MySQL commands.
Also you can see there are some entries without a User name (not only without a password). These databases are used for testing and benchmarking.
Ownership and permissions
Also you need to make sure the MySQL daemon runs under it’s own user, not some default (nobody) user. For this you must make sure that /etc/my.cnf lists this in the [Server] / [mysqld] (these 2 are synonims) section:
user=mysql
The container of the databases should be owned by the above user.
[root@i5 bin]# ls -ld /var/lib/mysql
drwxr-xr-x 4 mysql mysql 4096 Dec 1 14:51 /var/lib/mysql
More (official) reference on securing the initial default MySQL installation.