Here are a few more notes I ran accross from my former coach.
Assuming White has the pawn:
If the White King is behind the pawn, the Black King in front, the game can not be won. The key for Black is to keep his King in front of the pawn, and when he can not do so, in front of the king. So in a position where the King and pawn of White are side by side, Black should keep his King in front of the Pawn, one square away.
The key for White to win is to position his King in front of his pawn with at least 1 rank in between. Advance the king as far as possible without endangering the Pawn, and advance the Pawn only when necessary to protect it.
King and pawn vs. King while the pawn is on the edge of the board is a stalemate.
King and Several Pawns
When marching pawns, it is best to put your King beside your Pawns, generally one square ahead. Counting moves -- moving your King to encourage or force your opponent to move his -- can be very helpful.