To sign up for Google Docs, just go to http://docs.google.com. If you don't already have a Google Account, click Get Started to sign up for one.
Additional info on Google Accounts
A Google Account is made up of an email address and password of your choosing. If you already have a Gmail account, it comes with a built-in Google Account. Once you have a Google Account, just go to the Google Docs homepage and sign in.
The Google Account acts as a secure authentication method to keep your online accounts private while also allowing you to access numerous Google services with a single sign-in. Note that while it's possible to view a document, spreadsheet or presentation without a Google Account (via an invitation), you'll need to create a Google Account to edit the document, spreadsheet or presentation, and to continue accessing Google Docs.
While a Gmail Account is a Google Account, a Google Account is not necessarily a Gmail account.
Google Apps users, your administrator provisions accounts for your domain. In this case, you will access Google Docs using your Google Apps account at 'http://docs.google.com/a/yourdomain.com', please be sure to replace 'yourdomain.com' with your domain name.