How to keep apps logged in after changing phones

Hi guys - just purchased an XS ... my Iphone 6 phone was getting too slow :)

Backed up the phone, and switched - no problems at all - BUT ... each app was logged out and had to change some passwords since i forgot them!

An issue i had was with Messenger App - I had various accounts which I was logged on to - but when i opened the new phone, they were gone. They're still there on the old phone, but i need the app transferred to the new phone , with the accounts all there and logged in! Can this be done? I cant log back into them easily, as they are work related accounts, and some can't get the passwords, or reset them either (boss issues)

Thanks :D