They can all talk to each other and to the internet -- no one machine has to be the server. You simply connect the router to the modem and use the ethernet ports or wireless to link the netbooks to the router.
Make sure you specify what your present setup is when buying the router -- e.g. Cable or DSL.
If DSL, it may be simpler to buy a combined DSL modem/wireless router which is the commonest and probably cheapest kind and which would then replace the DSL modem.
So long as the network is operating reliably, all the machines have had their firewalls off at connection and filesharing is enabled on relevant folders -- it's mainly a matter of allowing the networking part of the game to operate.
I suspect the most you'll have to do is make an accurate note of the computer names -- or worst case run Windows' Add New Network Place for each computer (not an amusing prospect).