Is it possible to save a copy of a PDF form created in Livecycle Designer ES3?

You can do this with LiveCycle Reader Extensions, you can set a right to allow offline save.  You can also do it with Acrobat in a restricted sense where you have a limited number of submissions.

I cannot find the step by step process for reader extensions. It keeps telling me to use the Administrative Console, but I can't find it anywhere...help?

If you use the File > Distribute Form this does allow recipients to save data entered into the form.  However, beware of the following points:

1.  File > Distribute Form will take you through a process to gather information about your preferences for distribution including destination email addresses.  Once you have gone through this process, subsequently there is no option to change or reset any of the original inputs.  For example, if you wish to change the destination email address, you need to open the PDF in Acrobat Pro and go to Edit > Preferences and select the identity category.  The email address can be changed there.

2.The File > Distribute Form, seems to make other changes to the PDF too, because it uses the email address entered when Distribute Form is first used, to submit the PDF by email.  If you have an email submit button in your form, like I did, this does not work any longer in Adobe Reader.  Clicking it now simply prompts to save a local copy of the PDF. The submit does work in Acrobat Pro, but this is useless for a distributed form.

Paul