Hey Brian,
You're on the right track! Once you add a field to the customer table, you still need to display it on a Screen (for example New Customer or Edit Customer screen). To do this you must go to Customize -> Screen and then copy the screen you want to change.
From there you use Screen Designer to add the newly created field to the screen. For more details on how to use the Screen Designer check out the Screen Designer PDF
Additonally, you can also take advantage of the Webinars related to customization here.
Sam