Linux is a very secure system, however is as secure as its administrator is aware of security.
These are the most important tasks to keep a Linux distribution secure:
Only install software from the distribution repositories (apt-get, yum...). Avoid downloading pre-packaged (.deb, .rpm) software unless it comes from very known sources.
Learn to use IP Tables and create a configuration that let pass through your system only the traffic that you want to have.
Keep the system updated. The distribution repositories are updated as soon as possible when a vulnerability is discovered.
Understand user permissions and be as restrictive as possible.
Disable SSH passwords and enable login using SSH keys.
Enforce the use of strong passwords.
Disable the root account, so whoever needs root permissions will use sudo.
Install and learn to use SELinux. It will increase the basic Linux security.