It's awesome that you review the logs, many folks don't bother anymore.
It can be hard to grok it all though, and you might benefit from some tools that help you determine what's important and not (and only bother you when they see something important!).
As Joe mentioned, Logwatch is used by default on RHEL/CentOS. Logcheck is kept pretty up to date for Debian-based systems (though I don't know what they use by default).
I've also begun to hear some good things about Splunk:
http://www.splunk.com/Splunk is not free as in speech, it is free as in beer if you have less than 500MB of logs a day.
There's also host-based intrusion detection systems, like OSSEC and rkhunter:
http://www.ossec.net/ http://rkhunter.sourceforge.net/I guess where I'm going with all this is that it's really hard to keep up with the logs, know what's a hack attempt versus a legitimate connection, or just have time at all to browse them.
There's a number of excellent tools out there designed to help with exactly that though (the above is just a few examples!). I'd encourage you to keep trying to understand what all you're seeing. There's just too much though, you'll never be able to keep up with it, so at the same time I'd also encourage you to use automated tools to help you monitor the logs and keep the bad guys out.
-Eric
