How do I get my contacts back from the cloud after I made a new Apple ID account

I'm not an iCloud expert but I think not so easy or impossible. iCloud storage is tied to a particular ID. If you use Contacts on a computer you'd have to change the Apple ID settings for iCloud on the computer and then on the device to re-link to the Contacts database. This is why Apple says don't use more than one ID.

Create and start using an Apple ID - https://support.apple.com/HT203993

I believe that it can be done, in theory, by signing in on the old Apple ID with a Mac, moving all of the contacts to 'On My Mac,' signing out of the old Apple ID, signing in with the new one, and then moving the contacts from 'On My Mac' to 'iCloud.'

See Winston Churchill's response in the below discussion

How to transfer all files from one iCloud account to another

If you made a new Apple ID account, your contacts won't be associated with that Apple ID; they'll be associated with your old Apple ID. Why did you make a new account?