Since TC is a router by default, you will create all kinds of havoc if you just plug it in to the network.
Assuming the WiFi SSID and password are different on the new TC, power the old up with no network connections and edit it in Airport Utility. Set Network Mode to Off (Bridged Mode) and set WiFi to Off. Save the settings (will cause it to reboot), and plug the WAN port on the old to a LAN port on the new.
If WiFi is the same, change new WiFi to something different so you can distinguish which WiFi to connect to. After you are done, you can change it back.
Now, in Finder, navigate to the old one, copy all files to the new one.