How to add new email and new phone number to my apple account

Are you trying to change your Apple ID email address? Note the restrictions in the article.

Apple ID Name Change.  * 

 What to do after you change your Apple ID or password - Apple Support  *  I suggest doing this before you change your Apple ID or password.

or add an email to Reachable At, a Rescue or Notification email, address.

Apple ID email addresses.

For the phone number, you need to change/add the number on your Apple ID. You may have to add the new number before you can remove the old number. After the change, allow up to 24 hours for the change to sync. Before trying to make the change, sign out of Messages/FaceTime on all devices/computers.

Apple ID - Manage