The easiest approach is the Meitetsu Inuyama line from Nagoya Station (Meitetsu Nagoya Station) directly to Inuyama-Yuen Station, a 30-minute journey. Inuyama-Yuen is the closest station to the castle — it is a five-minute walk from the station to the castle gate. Inuyama Station (one stop earlier) serves the town centre and is the better option if you are heading to Meiji-mura or Little World.
JR Pass holders can take the Tokaido Line from Nagoya to Gifu, then the JR Takayama Line to Unuma Station, and walk across the Kiso River bridge into Inuyama — about 20 minutes on foot. This works but is slower and more complicated than the direct Meitetsu option. There is no direct Shinkansen access; Nagoya is the nearest Shinkansen station.