error: 'Access denied for user 'root'

[root@localhost mysql]# mysqladmin -h 127.0.0.1 -u root password "root123"
mysqladmin: connect to server at '127.0.0.1' failed
error: 'Access denied for user 'root'@'localhost' (using password: NO)'
[root@localhost mysql]#

[root@localhost mysql]# service mysql stop
Shutting down MySQL..                                      [  OK  ]
[root@localhost mysql]#
[root@localhost mysql]#
[root@localhost mysql]# mysqld_safe --user=mysql --skip-grant-tables --skip-networking &
[1] 30750
[root@localhost mysql]# 130606 13:10:20 mysqld_safe Logging to '/var/lib/mysql/localhost.localdomain.err'.
130606 13:10:21 mysqld_safe Starting mysqld daemon with databases from /var/lib/mysql

[root@localhost mysql]#
[root@localhost mysql]# mysql -u root mysql
Reading table information for completion of table and column names
You can turn off this feature to get a quicker startup with -A

Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 1
Server version: 5.6.12 MySQL Community Server (GPL)

Copyright (c) 2000, 2013, Oracle and/or its affiliates. All rights reserved.

Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

mysql> UPDATE user SET Password=PASSWORD('root123') where USER='root';
Query OK, 4 rows affected (0.16 sec)
Rows matched: 4  Changed: 4  Warnings: 0

mysql> FLUSH PRIVILEGES;
Query OK, 0 rows affected (0.00 sec)

mysql> quit
Bye
[root@localhost mysql]# service mysql start
Starting MySQL                                             [  OK  ]
[root@localhost mysql]# mysql -u root
ERROR 1045 (28000): Access denied for user 'root'@'localhost' (using password: NO)
[root@localhost mysql]# mysql -u root -p
Enter password:
Welcome to the MySQL monitor.  Commands end with ; or \g.
Your MySQL connection id is 3
Server version: 5.6.12

Copyright (c) 2000, 2013, Oracle and/or its affiliates. All rights reserved.

Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.

Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.

mysql> show databases;
ERROR 1820 (HY000): You must SET PASSWORD before executing this statement
mysql> SET PASSWORD FOR 'root'@'localhost' = PASSWORD('root123');
Query OK, 0 rows affected (0.00 sec)

mysql> show databases;
+--------------------+
| Database           |
+--------------------+
| information_schema |
| mysql              |
| performance_schema |
| test               |
+--------------------+
4 rows in set (0.00 sec)

mysql>