Firstly, there is a difference between the mysql root user and the system root user. Don't run the mysql client as root.
If you've never given msql's root user a password use:
Code:
mysqladmin -u root password NEWPASSWORD
If the root account has a password, you need to enter the old password to authenticate:
Code:
mysqladmin -u root -p'oldpassword' password NEWPASSWORD