How should I orient the chess board?
A white square should always be on the far right of the row nearest you.
In what order do I set up the back row of pieces?
From outside in: rook, knight, bishop, and then King/Queen. The Queen goes on the square that matches her color.
What's the proper way to refer to the location of a chess piece?
First say it's rank (row), then it's file (column). The rank will be a number and the file will be a letter.
If a pawn gets to the opposite side of the board and receives a promotion, can I make it any piece I want or only one that has already been captured?
Any piece you want. This means you could theoretically have nine queens.
What happens if it's my turn, I am not in check, but there are no spaces where I can move that will not put me in check?
This situation (having no legal moves) is called a stalemate. A stalemate is considered a tie.
Can I castle out of check?
No. There's no way to explain this; it's just the rules.
Can I castle through check (one of the spaces between where the king is now and where he will end up is threatened by an opponent's piece)?
No. Again, this is arbitrary but it's the rules.
Is a rook more valuable than a bishop or a knight?
Yes. A rook is informally given a value of 5 points, whereas bishops and knights are considered to be worth 3 points.
Which is more valuable, a knight or a bishop?
There's some disagreement over this. Both pieces are informally valued at three points, but many experienced players consider bishops to be slightly more valuable than knights.
Which squares are considered the most important?
Generally, the center squares (d4, d5, e4, and e5) are the best ones to control.
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