I just went through this frustrating exercise last week.
1. Back up your old phone.
2. Be prepared to have Apple force you to change your Apple password.
3. If the IOS on your old phone is newer than the new phone IOS (can happen) you will be instructed to set up the new phone as a new phone first, and to upgrade the IOS prior to restoring from the old phone backup.
4. Sign out of Find Iphone if you are signed in.
5. You may be prompted for a password to restore your new phone from the old phone backup. This is not your Apple password. This is a separate password you had to generate when you backed up your old phone at some point in time, probably the first time you did it, and you probably forgot it. Hope you didn't.
I did forget mine. I entered a "standard" password I might have used and thank god it worked. It took me all afternoon and lots of &%(#*$^ to get this done.