Pawns and the Castled King
Logical Chess Move by Move, by Irving Chernev, spends its first hundred or so pages addressing the importance of NOT moving your pawns from in front of your castled King because it creates weaknesses which are relatively easy to exploit.
But plenty of positions call for advancing these pawns. The most common are probably to make room for a fianchettoed Bishop, or to give the King luft in order to prevent a back rank mate.
I asked on /r/chess whether or not a fianchettoed bishop created any weaknesses similar to that of a pushed pawn. One response I got was,
"Having the three pawns in their original squares can also be a weakness, depending on the situation (esp. if a back rank mate is threatened). Generally, pieces defended only by the king are weak. The bigger the piece the worse defender it makes.
The fianchettoed bishop usually doesn't need much defence if it has mobility (a long unblocked diagonal to move) which makes it strong offensively and not a liability defensively. It can also defend the weak squares from almost every enemy piece rather well. Obviously if a pawn storm comes in then it's trouble."
In addition, when creating Luft, it is important to consider which pawn to move. There is such a thing as "strong luft" and "weak luft".
Obviously, a multitude of positional concerns should be taken into account, but all things being equal, it is usually better to move the h-pawn (or the a-pawn if the king is on the queenside) because moving the f-pawn can weaken the king's position and moving the g-pawn creates holes at f3 and h3 (or f6 and h6 for Black on the kingside). In the diagram, Black has a weak luft because of the holes on a6 and c6; White has a strong luft, without holes.