Do I need to buy any additional hardware to set up a wireless network for a printer - or is it just a matter of adjusting settings on the computer? I've found out how to share the printer, but I don't know what to do next.

Your printer needs to be hardwired into one of your systems and shared. Any other pcs that are on that same network (wether they are wireless or not) should be able to access that printer once you have gone through the "Add Printer" wizard and added the printer to "each" one. If you dont have the other pcs set up on an actual home network you need to do that or they cant share it. Those other pcs have to be able to log into that network to even see the printer.