You could use "taking" for a class/course. "While taking an English class..." You're referring to the period of time after the class started and before it ended. You could use "going through" for training, like the first sentence you have. This indicates the same idea: you're talking about after the training began and before it ended. They basically mean the same thing, it just sounds more natural with the pairs I showed here. I hope this helps!

I would say it like this : While going through training in the military, I fractured the back of my left hand.