do i get a new private key for every transaction that I transfer to my cold storage If I buy crypto in little bits at a time and I transfer to the cold storage wallet every time I buy some, would a get different private keys for each transfer? how to keep track of all those keys?

For example: If I buy 10 cryptocoins today and I store in my hardware wallet and I get private key A, then I buy 15 cryptocoins next week and I again transfer it to my hardware wallet, will I get private key B for the second transaction?
Thanks
Eve
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.