I've read that Ubuntu will support web applications out of the box, how can I set this up on my computer to play with it?
The webapps integration is included in Ubuntu 12.10 and newer. When you visit sites in Firefox or Chromium that have the integration enabled you will be prompted to add the site as an application, here's a list of sites that have support:
For fully integrated sites like google docs, you should see things like Dash integration:
After the site has been added you can launch it from the dash, or add it to your launcher, just like other applications.
The project homepage (where you can file bug reports) is here: https://launchpad.net/webapps
Related Topics and References: