Flights

To Tokyo (Haneda/Narita)
Zip Air: Cheapest from West Coast (esp. Vancouver)
Japan Airlines and ANA are also good choices. Avoid US/Canadian airlines if you can.
Remember to account for Bullet Train ("Shinkansen") to Nagoya (¥13000/person each way)
Flying to Nagoya
Requires layover, no direct flights from North America, but good to avoid needing to take Bullet Train.

Airlines with connections to Nagoya:
Chubu Centrair has a convenient train to get to Nagoya Station, where you can catch the train to Gero.
Trains

If starting from Tokyo (or another major city):
Take the Shinkansen to Nagoya Station(one of the major stops on the Tokaido Line)
Nozomi trains are the fastest. You may book reserved, non-reserved (cheapest), and "Green Car" (first class, more expensive) tickets at the departure station.
Train to Gero from Nagoya Station
Take the JR Hida line to Gero, which takes approx. an hour and a half.
Tickets cost ¥4500 each way for reserved seats, and the train typically leaves once an hour, but with a few gaps (be sure to check schedule). You may buy the tickets at any JR kiosk in Nagoya Station (they have an English option).
Getting to Yunoshimakan from Gero Station
Yunoshimakan has a shuttle that arrives at the station with every train arrival. It's purple and the driver will be holding a sign with "Yunoshimakan" written on it. They will ask your name on the room booking and will drive you to the venue.

It is within walking distance, but it's a steep climb and inconvenient with luggage. However, if you arrive early in the day you may leave your baggage with the front desk and walk down the mountain to the town to explore and have lunch before check-in at 3pm.