So that's where you got those damn screws...
Never really understood the card game so this all makes sense now.
The card game, and the various mechanics, is complicated...
But for the most part in the starting area, you get the Ifrit card, and try to win card battles from there. The card rules in Balamb are easy, you just have to have the highest number on the sides of the card, the highest number wins. GFs have skills you can learn that give you new abilities related to cards. This is where I get excited, because basically at the start of the game, using the card game mechanics, you can become a god by creating some of the most powerful magics in the game (though it's better to wait until after the first mission (dollet) as you get new GFS with new abilities).
You can create lots of items as well, including screws.
There are also ways to get better cards right away at the start of the game. For example, I just started a new game and my hand for winning cards was 2 Shumi Tribes, Krysta, Quistis, and Ifrit. All available before leaving the starting area. You have to really screw up to lose with cards that good.