They did not have enough training in such situations. The captain seemed more competent, but he was not on the controls to do the right thing. It's the same situation when untrained people drive cars to their limit. If the car slips then you should slightly steer in the direction of the slippage which is most of the time opposite to where the road turns. If you're not trained to do that you won't do it.

Do you really need training to say "I'm pushing the stick backwards now", "what are you doing over there?" etc

Surely that sort of basic communication about what you're doing should be one of the first things you learn...

That too. I just saw a BBC video about this incident and a pilot was saying exactly that - that the pilots should know who is in control and he should pilot the plane to the safety.

But I was thinking about training more in another sense. If your instinct tells you to do something that is opposite of the right thing than you should receive more training to brake your instinct. In this case the right thing to do was probably to push the nose down to gain speed. But if your instinct tells you that going down is not the right thing you won't do it. You should make a concious decision and you are not in the position to do that without prior exposure to the situation.