I gave someone an old phone and their Apple ID has taken over my new phone

I am not really understanding your issue. Nobody can sign into your phone remotely and override your Apple ID. More likely is you didn't set up the old phone properly for a new owner and they are using your ID. Even then you would have had to have given them various bits of private information in order for them to have been able to change your password.

Use the steps in this support article to remove your personal information from a device --> What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support