What's the easiest way to migrate all my files from my old Box account to my new one?

Hi Naomi, 

Welcome to Box Community and glad to assist! 

To move your Box files from your old Box account to the new one. You could do the Transferring of Folder Ownership. To do so, please make sure that the destination account is listed as one of your collaborators in the folder.  

Transferring the Folder Ownership: 

• Locate the appropriate folder
• Click the Collaboration Iconto the right of the file name.
• Alternatively, you can open More Options > More Actions and select Manage Collaborators.
• Find the collaborator whose access level you would like to modify. You can search by name using the Filter Collaborators search box in the upper right.
• Under the Permissions column, click the user's access level to modify it. Select the appropriate access level from the dropdown that appears. If the access level is greyed out, you may not have the necessary permissions to modify that user's access level.

Note: The collaborator you designate will be the new "Owner" of the folder and your access level will be downgraded to "Editor". The contents of the folder will no longer count against your account's storage allowance.

For more information, please see this article. 

Thanks for posting!