How do I move all my account settings from an old Apple TV to a new one?
I'm not aware of any way to migrate account information from one Apple TV to another. 

You'll have to set it up as new: sign into iCloud, download + sign into apps, etc. It is a bit of a hassle but it doesn't take too long. 

I suggest using the iPhone app "Apple TV Remote" which enables you to type passwords via the iPhone keyboard. Also, Apple now offers single sign on for certain cable providers, you'll find this in "Settings" under "Accounts" I believe.