Hi Catucas,
The cheapest way to go to Zakynthos is by bus or by train.
I would recommend the bus as the train is slower in this area
The bus leaves from Stathmos Kiffissos.
It is not so far from Athens center but really difficult to access by public transportation.
The best is to take a bus from Athens airport to the center of Athens and then take a taxi to "Bus Station Kiffissos".
Have a look at
http://www.ktel-zakynthos.gr/en-external-routes.html