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.