You could pick up the emails on both computers, but if you use POP it may take some organising. You should have the option in both email programs to choose IMAP
POP: Your computer takes a copy and tells the server to delete it at that end. Now the other PC can't get that email.
IMAP: the emails stay on the server (somewhere out there on the internet) and you view them from either computer.
More info here
You could have a home network and all modern operating systems should be able to link up, but I'm not sure that helps you really. (though you will love being able to pass files back and forth)
For that you need a modem/router which you probably have anyway The box that plugs into the wall for the internet probably has sockets for more than one PC. If both PCs are connected then you just have to run the 'Network SetUp Wizard' on each PC which will help you pick a name for your network and activate it.
Not sure what PC World had in mind. Some people make a network and set up an email server of their own on one of them. That's a bit much just to solve your immidiate problem.
Oh one thing you can do with a network is control your PC from the laptop and vice versa so that would mean you could view the emails on the other one.