Well, we already have a training ground, of sorts. It's called the arena:) It's not exactly formal, but we can add an area dummies and guards and trainers etc.
Right, but
Garile did bring up the valid point that the arena isn't well suited for educating the general population as to the basics of self-defense and defending the city. The arena always struck me more as a place for higher-level fighters thinking "hey, lets go kill some monsters/people for the heck of it!" rather than as a government-run facility meant to help the population of Hydlaa fulfill their civic requirement of knowing how to hold a sword and hit something with it (which is what the training grounds would be for).
I would imagine that
Garile's government-run training grounds would look like a handful of dirt lots surrounded by low wooden fences, with dried-seaweed-stuffed dummies propped up on wooden poles and some dried-seaweed-stuffed targets for target practice. On the grounds there would be a storage shed and a small office (for record-keeping), staffed by an older, laid-back, retired general and his younger, aspiring, too-serious-about-his-job assistant. Every city would probably have some kind of training grounds, though, especially if the requirement [of being able to defend the city when necessary] applies to all cities in Yliakum, so naturally the character of the staff would vary depending on which city's training grounds they were staffing. (which brings me to my next point below ... )
The laws make the populations as a whole sound more militaristic then the vibe I get from walking around the cities and talking to NPCs.
It would make sense that Hydlaa would seem very non-militaristic despite the law in question, since Hydlaa is probably one of the safest towns in all of Yliakum. So it makes sense that the citizens (and the staff of the training grounds) wouldn't be very serious about discipline and bearing arms since they wouldn't have as much of a need for it. Compare that to smaller villages closer to the Bronze Doors, where the residents face a very real danger of being attacked by monsters that slip past the Bronze Door fortresses.