A related Instructable shows how to link two microcontrollers via the internet http://www.instructables.com/id/Worldwide-microcontroller-link-for-under-20/
Step 1: Gather the parts
Picaxe 08M chip available from many sources including Rev Ed http://www.rev-ed.co.uk/picaxe/ (UK), PH Anderson http://www.phanderson.com/ (USA) and Microzed http://www.microzed.com.au/ (Australia)
Protoboard, servo, microswitch, 9V battery, 4xAA batteries and holder, tag strip, 10k resistor, 22k resistor, 33uF 16V capacitor, 0.1uF capacitor, 7805L low power 5V regulator, 10k pot, wires (solid core telephone/data wire eg Cat5/6), 6V lightbulb, D9 female socket and cover, 2 metres of 3 (or 4) core data wire, battery clips
The above companies also sell USB to serial devices which are useful for laptops which don't have a serial port. It is worth noting that some USB to serial devices don't work as well as others and it is worth getting one from one of the above suppliers as they have been tested for use with picaxe chips. The one that is known to work is http://www.rev-ed.co.uk/docs/axe027.pdf Of course, if your computer has a serial port (or an old serial port card) then this won't be an issue.
Step 2: Download and install some software
VB.Net (Visual Basic Express) is available from http://msdn2.microsoft.com/en-us/express/aa718406.aspx
If this link does not work then search in Google for: visual basic express download
The picaxe software is available from http://www.rev-ed.co.uk/picaxe/
You will need to register with microsoft to get the download - if this is a problem use a fake email or something. I actually found it helpful giving my real email as they send occasional updates.