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.