The official chess rules do not determine that the white goes first. Instead, this decision is left to the tournament-specific rules in the case of non-competitive play, in which case some kind of optional choice is often employed.

Basic Movement

Each chess piece has its individual style of moving. Moves are made to empty squares except when capturing an opponent's piece.

The king can move only one square horizontally, vertically, or diagonally. At most once in every game, each king is allowed to make a special move, known as castling.
The rook moves any number of vacant squares vertically or horizontally. 
The bishop moves any number of vacant squares in all diagonal direction.
The queen can move any number of vacant squares diagonally, horizontally, or vertically.
The knight moves to the nearest square not on the same rank, file, or diagonal.  Its move cannot be blocked by any other pieces, for example a knight with either move in a "L" or "7" shape.
Pawns complex rules of movements are:
A pawn can move forward one square, if that square is unoccupied. If it has not yet moved, the pawn has the option of moving two squares forward provided that both squares in front of the pawn are unoccupied. A pawn cannot make any backward movement.
They can capture an enemies piece on either of the two spaces adjacent to the space in front of them but cannot move to these spaces if they are vacant.

With an exception of the knight, pieces cannot jump over each other. When a piece is captured, the attacking piece replaces the enemies piece from its square. The captured piece is then removed from the game and this piece may not be returned to play for the remainder of the game.The king can be put into check but cannot be captured by any chess pieces.