Your setup should be:- Cable connection to modem-Wired connection to router-either wired or wireless from router to other devices (PC Laptop etc).
When to come to swap it's easy. Just unplug the old one from the router and plug the new one in (or setup wireless) and away you go.
If you do not yet use wireless you need to set it up with a wired connection.
Kymmy got in while I was replying