Considering the many flavors of linux, we're going to go over just the basics. Consult the documentation for your specific variation.
Set good passwords on all accounts (5 min)
Passwords are the first and best line of defense in protecting any computer. Select a good password on all accounts and remember remember them, especially the superuser/root/uid 0 accounts.
Use shadow tables and md5 encryption
Most installations use shadow tables by default. If you're queried, always use them. If they give you a choice of encryption schemes, of either 3DES or MD5, use MD5. MD5 is much more difficult to break.
Use ssh and sftp.
ssh and sftp encrypt the traffic that is sent back and forth over the internet. Its similar to putting your credit card number in a security envelope. Telnet and ftp do not encrypt anything and is analagous to writing your SSN or credit card number on a post card and mailing it across the country.
Use iptables or ipchains
Most distributions allow a user to select a security level or some sort of firewall. Generally, selecting "Medium" is reasonably sufficient. If you are planning on ssh'ing or sftp'ing into your machine, be sure to allow incoming connections to port 22.
Keep up to date with patches & security releases
Read Linux security documentation and implement appropriate changes.