Does my iCloud storage plan automatically transfer over when I change or buy a new phone?

The storage plan is tied to your Apple ID, not a specific device.

"Does my iCloud storage plan automatically transfer over to my new phone???"

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Transferring Over to a New iPhone:

No. You'll have to log in to your Apple ID on the new device. It is not transferred over.

Important:

When ridding of your iPhone, perform as instructed here, first, including the backup and iCloud logout: What to do Before you Sell, Give Away, or Trade In your iPhone, iPad, or iPod touch - Apple Support