These instructions are for the New Google Sites. On a computer, open your site.
At the top right, click "More " (the three dots).
Click Duplicate this Site.
Enter a name that indicates that this is your Duplicated Site
Click ok.
Share the copy of your site to your personal (not school) Google account by clicking on the person's outline on the upper-right-hand side and inviting the your personal account.
Make your personal account the owner of the Copied Site (not just an editor).
Open your copied site in your personal Google account. Test the documents. Share any documents from your school account that do not open while viewing your personal account to your personal account.
You can also see here for more about transferring: https://support.google.com/accounts/answer/3024190?hl=en