Yeah, that 'punch to pass' is called a handball, that's how you pass the ball to another player, if you throw the ball it is a foul.
If you kick the ball to another player and it is caught, then it is a mark, and the player (regardless of the team) gets a free-kick or they can choose to play on.
To run with the ball further than about 6 paces you have to 'dribble' or bounce the ball on the ground - otherwise it's a foul meaning you have run-with-the-ball.
To score you need to kick the ball between the goal posts as follows:
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if the ball goes through the middle posts without being touched by hand then it is a goal worth 6

. if the ball goes though either side, or is touched by hand, or hits a post it is 1 point.