Is there a website I can log into that will can check various exploits on my system, and basically tell me if it's patched up-to-date?
I think you would be better of by installing Nessus on your own computer. Nessus is free for home use and have a commercial version available too, and it lets you actively probe servers with a wide range of plugins.
Check it out: http://www.nessus.org/nessus/
Also.. could you trust a public webpage that probed your computer for errors? Setting up such a server would probably mean lots of problems for the host, since it would trigger IDS-alerts from other network providers. And you have no guarantee that they won't abuse their findings.
EDIT: Didn't know that they had changed the license on the plugin-feeds. Thanks K. Brian Kelley. Still worth it though. Superb software.
Qualys (http://www.qualys.com) provides a service that does this, but as with my comment to Commander Keen, it's not free. Qualys is generally recognized, too, having spoken with several IDS/IPS providers who also provide security monitoring services. The advantage with Qualys is you can get an appliance for inside your network, but you can also schedule a scan from one of their devices on the Internet.
There are also solutions like McAfee Secure (formerly HackerSafe) which provide those type of scanning services automatically.