Passwords and Credentials

Use different passwords for every site and account you have.
Use passwords that cannot be guessed, google "password generator".
Do not give out your passwords to anyone, especially people claiming to be site staff.
Do not log in to a site with another site's credentials, ever. For example, do not enter your Runechat username/password on another site. If the URL in the browser doesn't start with "runechat.com", it's not Runechat. Same goes for any other site.

General Internet Safety

Do not transfer valuables (GP, in-game items, BTC, $, whatever) to people unless you're really sure about the risks.
When you meet someone on social media who claims to know you, make sure it actually is that person. For example, to make sure you're talking to a Runechat admin or mod, use a ticket code first to confirm. And even then, be suspicious if they behave strangely (because other people get hacked, too).
Email addresses, TeamSpeak usernames, everything can be forged. Always confirm the identity of your communication partner.

Install a personal firewall so your computer isn't exposed to the internet unprotected. OS X and Linux come with firewalls built-in, and it shouldn't be difficult to find something for Windows.
Especially if you're using peer-to-peer software like Skype or BitTorrent, use a firewall to shield your computer. Keep in mind that other users on these networks can see your IP address and try to attack you.
Don't install software from dubious sources, because it could contain Trojans and other malware. If you can't help it, do a virus scan on it first, but keep in mind that virus scanners are not 100% effective.
If you ever think your computer itself might be compromised, wipe the hard drive and do a clean re-install of the operating system.
Regularly back up your data, for example onto an external drive or use a cloud-based backup service. Back up your important online stuff, too.