From my limited experience:
If kings are castled on opposite sides, you usually have to your push pawns to his castle too, it becomes sort of a race.
If the kings are castled on the same side and he's pushing kingside pawns, remember to stay calm, it usually looks more dangerous than it really is, remember they're weakening their king's position.
I realize these are not very helpful, but rules of thumb are usually very limited without example positions.
I have been improving my game, but once in awhile I will play a very inexperienced player and end up losing because I can't handle the pawn rush. I was wondering if there are some rules of thumb to follow.