Scientific Linux was designed so that an administrator can easily upgrade their computer between minor releases.
It is not required that administrators upgrade their release, because the security errata will be updated for all releases. But there are often features and/or packages included in later releases that were not available with the earlier releases.
Below is the steps, using yum, to change to the latest 6.x release.
These instructions are for updating between minor releases, such as from 6.0 to 6.1. Changes between major releases 5 and 6 (such as from 5.2 to 6.1), cannot be done with yum, but must be done with the main installer.

YUM

For The Impatient

1. yum clean all
2. yum --releasever=6x update
   Note: 6x can be replaced with whatever version you wish to go to, such as 6.2, for example yum --releasever=6.2 update
3. Updating from 6.0 only yum install yum-conf-sl-other

For those a little more cautious

1. Make sure you have done your backup's.
   I haven't heard of anyone needing them, but don't you be the first.
2. Make sure you have enough disk space. In particular, make sure you have enough space in /var/cache, which is where yum downloads the rpms.
3. optional yum update
   This will get your system up to date with the latest packages. It will also make the last step smaller.
4. yum clean all
   This cleans up all the headers and packages from your old yum configuration. If you do not do this step, it is very likely that yum will say that there is nothing to do.
5. yum --releasever=6x update sl-release
   Note: 6x can be replaced with whatever version you wish to go to, such as 6.2, for example yum --releasever=6.2 update
   This will install the latest sl-release your system. sl-release tells your system what version of SL it is.
6. Updating from 6.0 only yum install yum-conf-sl-other
   In SL 6.0, 'security' and 'fastbugs' were both in the sl-updates.repo, provided by the sl-release rpm. In SL 6.1, 'security' is now in the sl.repo, provided by the sl-release rpm, and 'fastbugs' in the sl-other.repo, provided by the yum-conf-sl-other rpm.
7. yum update
   This is the big part, where everything gets changed.
8. Check your grub settings. Although yum usually get's things right when it updates kernels, it is always a good idea to give the grub config file a check.
9. optional yum clean all
   This cleans up all the rpms that you just downloaded, clearing up your disk space.
10. /sbin/reboot
    Reboot into your updated Scientific Linux release