Network "kits" often contain more than you need. Can you tell us what connectivity each PC has please? Your question "do I need a card in each PC?" suggests that you may not have any connectivity devices or at least may not know whether you do.
You'll need an Ethernet - normally either 10/100 or Gigabit speed - to configure the router. Note that this MAY not apply to every single router, but it is normal and is for obvious security reasons - until you enable a security protocol, the world and his wife can enter your network if they are in range and a router that opens its config page to a wireless computer is foolish.
You don't HAVE to run the router with one PC wired to it once it's set up, though it may be argued that this is more reliable should the wi-fi element be problematic.
You don't need anything new to make the printer network-ready unless you wish to use it while its host PC is switched off.