Train Travel to Zao
The most common train station for access to Zao Onsen is Yamagata Station.
From Yamagata station, you can catch a bus to Zao Onsen Ski Resort which takes around 40 minutes. No reservation is required for this bus trip from Yamagata station to Zao Onsen town, but the trip will cost around 1,000 yen for adults and 500 yen for children.
From Tokyo: the bullet train is the best way to get to Zao, with bullet trains titled "Tsubasa" running directly to Yamagata station in under 3 hours. Other bullet trains will take you as far as Sendai station, where you will then have to transfer to a local train for the remaining 1:30 hr of the trip.
From Osaka, Kyoto and Nagoya: travel to Tokyo by bullet train before following above instructions.
Search Japan's train schedules here: HyperDia
Japan Rail Pass: For travellers visiting cities either side of the ski resort stay, a Japan Rail Pass is a fantastic investment, allowing unlimited train travel on most Japan Rail trains, for periods of 7, 14 or 21 days to suit your travel schedule.