do i get a new private key for every transaction that I transfer to my cold storage
Every address you use to receive crypto will have a corresponding private key. So if you buy 10 coins today and transfer to your wallet, it will have a private key for the address you use. Next day, you buy 15 and transfer to a different address generated by your hardware wallet, it will have a different private key.

You do not have to keep track of the private keys. All those private keys are generated from the same root key which is generated from the 12/18/24 word seed you set up for the hardware wallet. All you need to remember/record is the 12/18/24 word seed.

Try playing around with this https://iancoleman.io/bip39/ . It will give you an idea of what happens inside the hardware wallet.