I have a new Apple Watch 8, what process do I need to do to replace old Apple Watch with cellular & connect new Apple Watch with cellular to both Verizon & Apple my Apple ID?

First unpair your old watch to slso remove data and activation lock

Unpair and erase your Apple Watch – Apple Support (UK)

(this will create a backup too)

Set up your Apple Watch

Then set up your new watch, accepting to restore the backup

Set up your Apple Watch – Apple Support (UK)

which includes this article how to set up cellular

Set up mobile on Apple Watch – Apple Support (UK)

 As I am outside USA I am not sure if Verizon need to be contacted first but you could check with them.