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This section is intended for those who have not previously dealt with the security implications of having a full-time Internet connection. Or may not understand some of the basic concepts of security. This is meant to be just a quick overview, not a comprehensive examination of all the issues! Just enough to give you a gentle shove in the right direction. Please see the Links section for sites with more details. Also, your distribution surely has plenty of good information as well.
Turn off any daemons and services that aren't absolutely essential, and can be accessed from outside. You can't get compromised through a port that isn't open. Use ps and netstat to see what services are running. (See man pages for specifics). Do you really need named, sendmail, telnet, ftp running and accessible to one and all? If not sure, then they should not be running. Then take whatever steps necessary to make sure they don't start again on the next boot. See your distribution's documentation on this.
Many distributions start some well known services by default. You may not have done anything yourself explicitly to start these. And may not even realize these are indeed running. But it is up to you to know what is running, and how safe it is. Don't rely on a "default" installation of any distribution to do this for you, or to be secure. Chances are it isn't.
If you decide some services are essential, make sure you are running the most current version. Exploits are found, and then get fixed quickly. Don't get caught with your pants down. A full-time connection makes staying updated very easy -- and very important. Check with your distribution to see what new packages are available. Then stay in touch. If they have a security mailing list, get on it.
Take passwords seriously, using non-dictionary "words". Use shadow passwords (this should be a standard feature of newer distributions). Do not allow remote root logins. See the Security HOWTO for more details and ideas.
Use ssh instead of telnet or rsh.
Set up a firewall to limit access, and log connection attempts. This will be different depending on which kernel series you are using: ipfwadm for 2.0, ipchains for 2.2, and iptables for 2.4. See the below HOWTOs for a more in depth discussion on this and other security related topics:
Security-Quickstart-HOWTO and for Redhat based distros Security-Quickstart-Redhat-HOWTO
Additional references are in the Links Section below.