It's not too complicated.
If you get a router with your BT internet, you can do most of it through that and won't need a separate network switch.
What you want is something like this:

The "ADSL Wireless Router" is your internet router. Connect this to your BT line (with an ADSL filter) and connect each of your computers to this, either with Cat5 cables or wirelessly.
The "NAS Drive" is your backup drive. NAS = Network Attached Storage. Eventually you'll probably want a proper server (a dedicated computer) for centrally storing files).
You'll also need a way to connect your printer to the network. You can either buy a printer that's wireless like this:
http://www.amazon.co.uk/P1102W-LaserJet-Wireless-Printer-AirPrint/dp/B003ANMPTI/ref=sr_1_14?ie=UTF8&qid=1323359187&sr=8-14or you could just connect a printer to one PC and then "share" it with the other computers.
When you are ready to upgrade, you'll probably end up with something like this:

Which is actually similar to what you originally suggested.
If you don't have enough network ports (sockets) on your internet router (most only have 4 or 5), then you will need a network switch connected to your router to add more ports.
Your questions says 3 x PCs, plus 1 printer and 1 backup drive. That's 5 ports used if you don't use wireless.