Schattenburg is the castle above Feldkirch and the town's essential sight. Its walls rise directly from the edge of the old centre, close enough that you can climb up between errands, but high enough to give a proper view over the rooftops and valley.
Inside, the museum covers local history, weapons, rooms, and the Montfort counts without overwhelming you. The restaurant is known locally for enormous schnitzel portions; it is touristy, yes, but not in a joyless way.