you would be better off if you upgraded the older PC to Windows XP - it will make your job a whole lot easier.
You'll need to install a PCI network card in your old machine if it doesn't already have one, and that's a pretty easy job. You simply open the computer's case, and insert the card into any spare PCI slot. Reboot the computer and Windows XP will recognise the card and install its own driver.
Once you've done that you can connect the computer to a spare Ethernet port on the router, and that's the hardware part over with. All you need to do now is run the networking wizard on both computers. Make sure that you enter the same workgroup name on both machines. You'll be offered MSHOME as a default, but you can choose another name if you wish. Each computer must have a unique computer name, and again you can choose any name you like.
Running a wireless connection would mean buying a wireless modem/router combination, plus a wireless network adapter for the older machine (you can get USB adapters), and there would seem to be little point in doing this if the computers are both close to the router. If one machine is in a another room it's a different matter - it would be helpful to go wireless in that situation.