IT ...

Set password for root,

mysqladmin password

Put in /etc/my.cnf (chmod 600), so it's convenient for admin, it's not a good practice, though.

[client]
user=root
password=newpasswd

For individual database, grant rights to its owner,

grant all on database1.* to 'user1'@'%' identified by 'password';
flush privileges;

If you did not record down password for root and forgot it, start mysqld with --skip-grant-tables

run "mysql mysql" with following statement,

UPDATE user SET Password=PASSWORD('YOUR_PASSWORD')
WHERE Host='localhost' AND User='root';
exit

The restart mysqld normally.