I have an old Apple TV ("2nd generation") and have just purchased a new one as a replacement. How do I move all my settings from the old one to the 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.