Let's look at all the ways to change MySQL password, for root and other users:

In MySQL the default password is empty. This is inherently unsafe and should be immediately changed. Here is how you can change MySQL default root password:

mysqladmin -u root password NEWPASSWORD

Here is how you can change OLDPASSWORD to NEWPASSWORD

mysqladmin -u LOGIN -p OLDPASSWORD NEWPASSWORD

Replace LOGIN with your login name, OLDPASSWORD with your current password and NEWPASSWORD with the new password.