To change the root password, we need to still start mySQL with –skip-grant-tables options and update the user table

1. Stop mysql:
systemctl stop mysqld

2. Set the mySQL environment option
systemctl set-environment MYSQLD_OPTS="--skip-grant-tables"

3. Start mysql usig the options you just set
systemctl start mysqld

4. Login as root
mysql -u root

5. Update the root user password with these mysql commands
mysql->UPDATE mysql.user SET authentication_string = PASSWORD('MyNewPassword')
    ->WHERE User = 'root' AND Host = 'localhost';
mysql-> FLUSH PRIVILEGES;
mysql-> quit

*** Edit ***
For 5.7.6 and later, you should use
   mysql->ALTER USER 'root'@'localhost' IDENTIFIED BY 'MyNewPass';
Or you'll get a warning

6. Stop mysql
systemctl stop mysqld

7. Unset the mySQL envitroment option so it starts normally next time
systemctl unset-environment MYSQLD_OPTS

8. Start mysql normally:
systemctl start mysqld

Try to login using your new password:
7. mysql -u root -p