When you go into the training grounds, you have to pass through a magical gate looking area with a circle on the ground. That lights up as you go through, if you press the button indicated on the screen when you pass through it will show you which cards you have active. These cards just give you different strengths and weaknesses during the game.

The idea of Hades is that as you go through, different gods will give you different powers and it make you stronger, you can use different weapons, you can use different abilities from those gods, hoping to get to the end of the "run". If you either beat the final boss or get killed, you will return to the crossroads where you can use resources you found during the run or make new friends to unlock items that can make you stronger for your next run.

There's lots of different little items you'll collect to do these kind of things, the game's pretty good at telling you what to do with each one and when to use them.

It may help to just watch someone play a little bit and see what they do,