Nagoya Castle, Toyota Commemorative Museum of Industry & Technology.

No so obvious things:

Tokugawa Art Museum

An impressive display of swords, armour, costumes from Noh drama, elegant traditional living, and various aspects of the scroll of the Tale of Genji (the original is too delicate to be on permanent display).
Address: 1017 Tokugawa-cho, Higashi-ku, Nagoya.
Take the JR Chuo line to Ozone station, 10mins walk from south exit.
Open 10:00-17:00. Closed Mondays.
Entrance ¥1,200

Robot Museum

In a central location in Sakae. All sorts of robots. The Japanese love them, and you (and any kids) will too.