I tried to explain as much as I could using illustrations. If the written directions are confusing, take a look at the pictures. For the more complicated rules, I illustrated them in sequence.
Step 1: Setup, Turns, and Taking Pieces
The board is setup as shown. There should always be a white square at the closest right-hand side for both players. Remember that the queen must be on a square that matches her color.
White always moves first, and players alternate turns. Players can only move one piece at a time, except when castling (explained later).
Players take pieces when they encounter an opponent in their movement path. Only pawns take differently than they move (explained later). Players cannot take or move through their own pieces.
Step 2: Pawn Movement
If a pawn reaches the opposite side of the board, it is promoted to a higher piece (except king). There is no limit to how many pawns can be promoted.
Step 4: Knight
To make it easier to remember how a knight moves think of an L. Two spaces in a direction forward, backward or side-to-side, and one space at a right turn.
Step 6: Queen
Step 7: King
A king cannot move to a square that is under attack by the opponent.