Here are my observations so far:
To get everyone to train properly:
1. Schedule a training order, with the minimum set to the number of dwarves in the squad.
2. Build TWO barracks type things (I used an armor stand and a weapon stand)
3. Define a barracks from each, overlapping with each other. (Not sure if that is necessary--I've only done this once)
4. Set both barracks to '[T]rain' the squad you want to train there.
5. Toggle the squad to inactive, unpause, then toggle to active/train (or whatever alert you have them scheduled to train in). I happened to have all labors disabled for my soldiers when I did this, but I doubt its necessary, as they all turned into the soldier graphic when I did this.
Before I did the above, I would have some of the dwarves deciding to train at the barracks in their civilian graphic, doing individual combat drills or something, and randomly some others who would turn into the soldier graphic and go to/organize group demonstrations. The ones who would turn into the soldier graphic seemed random, and would switch around every month (some months none would turn soldier, some months a few would. I had only done step 1, with 1 barracks set to train the squad.) As soon as I built the second barracks, set it to train the squad too, and toggled inactive/active, the entire squad turned soldier and went to train.
Also note that they will no longer starve or dehydrate themselves in 31.10, so you can safely have them on-duty year round (I have, and haven't had any problems with their emotional well-being)
Another problem I had was with training specific weapons--I noticed after a good while of training, half the squad hadn't gotten any weapon skill, and was still just doing striking/dodging. Looking in their inventory, they would have 2 shields, one in each hand, or sometimes just 1 shield, but no training axe (even though their uniform said to get one). I defined an empty uniform, told it to replace civilian clothes, and assigned it to all dwarves in the squad. After they ran and put most of their equipment away (for some reason some of them kept a shield), I swapped them back to the uniform with both shield & training axe, and they all went and got one (although some also had a shield... i.e., 2 shields and a training axe). Anyway, now they are all learning how to use axes