The start of the game and the process of each turn are things I'm covering very quickly because I want to show off the unique parts more. If you have other questions, though, I'll answer them.
To begin, each player takes a deck with a certain amount of cards, and both players draw 5 cards from the top. If a player doesn't have a card that costs 0 in their hand, s/he must reshuffle and then draw 4 cards. The game then begins no matter what was drawn.
During one player's turn, they start by drawing one card for free. (They can draw a second one at the cost of 10 points, and a third one for another 20 points.) Next, they place cards on their side of the grid, spending points if the cards have a cost. While doing this, they can also attack with their Field cards to gain points, or claim an Arrow Piece. When the player is done, it's the other player's turn to do the same thing.
As far as attacks and battling goes, this area is more interesting. Once per turn, any of your Field cards that are able to attack can do that. If they can attack in more than one direction, you have to choose which way you want them to attack. If the attack goes through blank spaces, or cards that don't affect the attack at all, you simply gain the number of points on that arrow.
These are the squares that Bob's attacks will go through if he's placed on the front row, on the space second to the left.
However, in a case where you're also facing an enemy's attack, you'll only earn points equal to how much greater your attack is - in this example, 10 points is reduced to 2. Also, the opponent's Field card may be knocked out and discarded, depending on its Defense.
Points can be further reduced if there is a Field card in the back row with Defense. The attack above is worth 4 points until it crosses the other card, which reduces the rest of the points. Also, if there's a card in the front row blocking the attack as well, the card in the back row cannot be discarded. However, you can just attack with the card on the right, where it's not protected.
The other use of Defense, and probably the most important one, is survival. If you go for an enemy card and have a higher attack than it from that direction, the enemy is only knocked out if it has less Defense than the amount you won by. If not, you still earn the points, but the card stays where it is. Sound weird? Look at the first image again. You won by 2, so if the enemy has 2 or more Defense, it still survives.
As you can see here, you have just made the enemy Field card useless... but you also can't actually get rid of it. You better find some other way to destroy it before one of your cards stopping it gets knocked out.
Weak Field cards don't cost anything; you can play as many as you want in one turn. The stronger ones cost points, though, so you need to earn some using the weaker ones first. Also, keep in mind that these are the same points you use to trade for those Arrow Pieces. Do you strike quickly with lots of weaker cards and try and get the Arrow Pieces fast, or is it too risky since the enemy can stop you with stronger cards? Do you set up an unbreakable wall of cards, even though it'll require a lot more total points to win? Hmmm...