OK, a router is an independant device that connects to the internet itself and therefore is not reliant upon the PC at all. All that you have to do is to program the router with the username and passwords applicable to your account and away you go - I would think this has been done for you when it was first set up.
Yes, basically all you need to do if you get a new PC is to install a network card, this would require drivers to work but then a new PC will have windows xp and therefore I suspect the drivers will be installed automatically.
Further, most new motherboards (and therefore PC's) come with ethernet connections on them already and so there is no need for a network card. Simply get a RJ11 cable and connect to the router and you will be away.