The only way I know of that you could do that, without connect them to a computer, would be to use the old phone over WiFi (doesn't need to be on an active account to use open or personal WiFi) then either upload the data from the old device to a cloud or similar account, or email it to yourself (that will take longer, email limits size). Then download it all to the new device, or save from emails on the new device.
To be honest though, doing it via a computer USB connection would be less of a hassle and way faster.