How can I migrate my foto data from my old Lightroom classic account (School) to the new (private)?

With Lightroom Classic it is simple. Just use the Creative Cloud account to sign in to your new private account before opening Lightroom Classic. 

It possibly depends on - where your data is stored.

If your photos and catalog remain in the same location (your computer?), then there is NOTHING to migrate, NOTHING to Import. You simply [Sign In] to the Creative Cloud Desktop App with your new account and carry on as before.

Can you provide more details of your system and file locations?