where does the image of my face go? How is it used?

Assuming you're talking about FaceID, Apple, understandably, don't give deep details of FaceID's inner working, but some parts are known.

At the very least, it doesn't keep a copy of the image of your face. No one, no matter who they were, would ever be able to see a picture of your face from your FaceID data (of course, if they get into your phone they may see photos of you in your photo library, but that's a different issue).

Instead of an image, FaceID builds a profile of facial dynamics - conceptually, think of it as things like metrics, shape, etc. - the distance between your eyes, the curve of your nose, the shape of your ears, etc. The specifics of which metrics it records are not public, but they're stored in an encrypted enclave on the device.

When you try to use FaceID to authenticate, the camera looks at your face, pulls out the metrics, and compares them to those stored in the database. If the database matches, the authentication passes.

Note also that there's no typical way to get at the metrics in the database to 'see' or rebuild an image of what face is recorded. Instead, FaceID builds a profile of whose looking at the camera and essentially says hey, I have a face here with these characteristics, does this match what's in the database?, with a simple yes/no response.

There is no “image” of your face, rather a mathematical contruct is created based on your face and it’s saved in the secure enclave of your device and never leaves.

Which is why you ALWAYS have to setup Face/Touch ID again when changing devices.

You can read more here:

Face ID and Touch ID security – Apple Support (AU)

About Face ID advanced technology - Apple Support

Advanced technologies

The technology that enables Face ID is some of the most advanced hardware and software that we’ve ever created. The TrueDepth camera captures accurate face data by projecting and analyzing thousands of invisible dots to create a depth map of your face and also captures an infrared image of your face. A portion of the neural engine of the A11, A12 Bionic, A12X Bionic, A13 Bionic, A14 Bionic, and A15 Bionic chip—protected within the Secure Enclave—transforms the depth map and infrared image into a mathematical representation and compares that representation to the enrolled facial data.

Security safeguards

Security is important to all of us to protect information on our devices. We have done some important things to safeguard your information, the same way we did with Touch ID. Face ID uses the TrueDepth camera and machine learning for a secure authentication solution. Face ID data—including mathematical representations of your face—is encrypted and protected with a key available only to the Secure Enclave.

Privacy

Privacy is incredibly important to Apple. Face ID data—including mathematical representations of your face—is encrypted and protected by the Secure Enclave. This data will be refined and updated as you use Face ID to improve your experience, including when you successfully authenticate. Face ID will also update this data when it detects a close match but a passcode is subsequently entered to unlock the device. Face ID data doesn’t leave your device and is never backed up to iCloud or anywhere else.

Setting up a new phone doesn't change anything about the old phone. But, if you plan on getting rid of the old phone, start here:

What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

Re: “… I set up a new phone, but my OLD phone still functions like before …”

This is INTENTIONAL.

If there is some problem w/ your new phone - or the data migration - you’ll still have your old phone to “fall back on.”

Once you’re satisfied that everything you expected / needed to transfer DID, then “wipe” your old phone as instructed in previous reply.