If there is no service on the one device, then there is no way to get the information off the phones internal storage without a computer to connect to.
Now, if you can connect to WiFi on the old device (that should still work, even if no service on the phone) then you can get yourself a cloud account, google account, etc., and copy all the files you wish to save to that account. Then turn around and download them all to the new device. Just be very careful to leave them all on the old device until you are positive they are on the new one as well. I have seen to many people delete things thinking it transferred when it didn't.