From your existing wordpress site (must be 3.0 or higher), you'll need to install the latest release of the Cambrian Explosion plugin. For convenience, cambrian.zip is pre-packaged to drop right into the Wordpress admin.
You'll also need to activate the plugin (for multisite, you should activate from whichever subsite you'll be exporting). There should now be a new item under the Tools menu.
Click the button and you should be presented with some output and eventually an export zip file.
You'll need to follow the guide for generating a new site, but stop once you reach the Wordpress install page — any configuration will come after the import process.
Run the import script in your newly generated site, provide the path to the export file when prompted, and follow any further prompts.
./bin/import
Once complete, open your local site in the browser, which should have all the content, themes, and plugins from your old site.
Some activation, reconfiguration, and cleanup may be necessary depending on your existing plugins and themes.
From here, you can continue to follow the guide for preparing and deploying to remote environments.