Functionality changes - so this data is really only relevant in and around late 2011.
I had a client with Google Apps that changed their company and domain name. As such, I created an entirely new account. I moved email, calendar, and contacts by downloading their email with the Google Apps Sync for Outlook, exporting to PST, and then uploading the data to the new mailbox.
But Google Docs wasn't so easy. You're supposed to be able to download all your Google Docs into a single ZIP file. But that wasn't working for me. So here is how I moved Google Docs from one Google Apps account to another that were in different domains.
On the old account, I used this utility to download the Google Docs into a single folder:
Now, I had all the docs in a single folder and they were all in the Word, Excel, PowerPoint, PDF, etc format.
Unlike their download function, Google's built in upload function was working fine. I clicked on the upload button and there were options for "file" and "folder." I chose the folder where I downloaded all the files using that utility above and it imported each Word file into a Google document and each Excel file into a Google spreadsheet, etc. I lost the collaboration information and the dates of the files, but I was able to keep the file content with relatively little trouble.