Does my apple care transfer over when I get a new iPhone or do I have to but it again?

No. Apple Care+ is device specific it does not transfer over. If you no longer need Apple Care+ on the old iPhone you can cancel it.

You can get Apple Care+ for the new device directly from the Settings App.

See here: Cancel or change the payment method for your AppleCare plan - Apple Support

If the phone is replaced under warranty, the replacement will be covered by the remaining Apple Care.