To use a custom domain with Google App Engine, you would follow the steps here: Google App Engine: Using a Custom Domain
When you get to step number 3 there, be sure to click:
Sign up for a free single user Google Apps account
After this, as written there:
"Proceed through the Google Apps signup process, either using an existing domain name or registering a new one."
Since you already have a domain name registered through GoDaddy, follow the steps for using an existing domain name.
Then you'll have to create a CNAME with GoDaddy to point your domain to the subdomain you created at
appspot.com. Here are instructions on how to do this: GoDaddy: Adding or Editing CNAME Records
You'll then have to wait 24-72 hours for DNS servers to update so that your domain name resolves properly.
For additional help, there is a tag on StackOverflow that you can add to questions about Google App Engine. See this for more on that: Google App Engine community support is moving to Stack Overflow
There is also a support forum for GoDaddy domain name issues here: GoDaddy: Domain Management and Services Topics
According to this post yesterday, the free, basic version of Google Apps is being discontinued, and replaced with a $50/yr per user fee that includes:
"24/7 phone support for any issue, a 25GB inbox, and a 99.9% uptime guarantee with no scheduled downtime"
This won't affect existing users already using the free service (or educational organizations). See this for more information: Changes to Google Apps for businesses
They will however offer: "A $50 App Engine credit for developers who need to purchase a new Google Apps for Business domain in order to associate a domain with an App Engine account."