On my system, you need root access to view the logs, so you may have been getting permissions errors if you tried to access the files under /var/log as an ordinary user instead of root. You can view the logs with any text editor, but it may help to use a log parsing tool like logwatch from
http://www.linuxpackages.net/pkg_details.php?id=4153.
I have mine configured to medium detail level and this as well as manual parsing of my logs is very useful.
As to what to look for - it's best to look in the logs first and then search the man pages for what to expect, then ask questions here about specific entries.