First, I would recommend to install some third party softwares to analize the logs files for you every early morning and send an email.
Read their documentation and install it at your own risk.
There are several good ones but do a search on google and you'll find out more.
Do a cat /etc/syslogd.conf, all your logs will be recorded according to the configuration inside the file syslogd.conf
Read them carefully and do a search on google when it's neccesery.
Configure iptables or ipchains to access only the ports that are considered being used. Or maybe install a firewall to do that for you.
No machines are completely secured and every of them brings the chance of being compromised. Make sure all your packages are up-to-date. If you think it's neccesery, either you have to do lots of reading or hire a sys admin dedicate to taking care your server.