"Chess is a sea in which a gnat may drink and an elephant may bathe"
The game of chess is unique in its complexity, history and heritage. For this reason there are many ways in which you and your children can approach and enjoy the game.
The game has its origins in India before the 6th century. The pieces we now know as the Pawn, the Knight, the Bishop and the Rook represented the four divisions of the Indian army: the infantry, the cavalry, the elephants and the chariots. The King and Queen were the ruler and his advisor. It is this, in an updated form, which forms the basis of the army that Sam and Alice join in the book.
Chess followed the trade routes from Asia to Europe, and, in Spain in about 1475, dramatic changes were made in the rules, leading to the game we know today. The powers of the Queen and Bishop were increased, and the two-step pawn move, castling and en passant were introduced.
Since then, the game has been studied extensively. Many thousands of books have been written on all aspects of the game. The computer revolution has totally changed the way the game is studied. Databases of up to 5 million games are available, and perused in detail by master standard players, as well as by many amateurs who aspire to reach that level. The game is so hard to master that even the best playrs in the world frequently fail to find the best moves, but at the same time it's simple enough for young children to have little problem mastering how the pieces move.
For many, perhaps most, who just want to enjoy the occasional game with their family and friends, you don't need to know much more than the rules, but if you want to play competitively, the more time you spend studying and practising the better you will get.
But, whatever your ambitions there is one basic principle you need to understand. Other primers ignore this because they think it obvious, but for most children it is anything but.
SUPERIOR FORCE WINS!
We give each piece a very approximate value: PAWN: 1 point, KNIGHT: 3 points, BISHOP: 3 points, ROOK: 5 points, QUEEN: 9 points. (This scale slightly undervalues the Queen and Bishop, but it's sufficient for beginners.) Other things being equal (which often they are not), an advantage of 1 point is, with best play, often enough to win, and an advantage of 2 or more points is, again with best play, almost always enough to win.
This means that a chess player has to do three things every move: see if s/he can play a move to gain points, see if the opponent is trying to gain points, and ensure that the move played is safe and does not lose points.
These are hard lessons to learn, but in the book Sam and Alice have to learn them in order to defeat the alien army and save the planet.