1. Change to root User

su -
## OR ##
sudo -i

 

2. Backup all important data

• Backup /etc diretory
• Backup important logs /var/log
• Backup web server configs and sites
• Dump MySQL databases
• Dump PostgreSQL databases
• Backup all what you need if something goes wrong

 

3. Check list of packages that are going to be updated

yum list updates

 

4. Upgrade with yum update

Official way to do upgrade:

yum update

Another way to do update is first clean all, second update glibc, yum, rpm and python packages and then update other packages like following:

yum clean all
yum update glibc* yum* rpm* python*
yum update

 

5. Reboot

reboot

 

6. Check CentOS 6.5 (Final) release info and Check your entire system

cat /etc/redhat-release
## Output ##
CentOS release 6.5 (Final)

Following needs redhat-lsb package

lsb_release -a
## Output ##
LSB Version:    :core-4.0-amd64:core-4.0-ia32:core-4.0-noarch:graphics-4.0-amd64:graphics-4.0-ia32:graphics-4.0-noarch:printing-4.0-amd64:printing-4.0-ia32:printing-4.0-noarch
Distributor ID: CentOS
Description:    CentOS release 6.5 (Final)
Release:        6.5
Codename:       Final

Check that your system is working normally and test every services.

 

Source: http://www.if-not-true-then-false.com/2011/upgrade-centos-6/