I recently borrowed a book from the library by Jeremy Silman, called the Complete Endgame Course. It has taught me an immense amount already, and I still have it for a few more weeks. There was one exercise in it which I thought I would show you.
This is an odd task, but it can prove very useful to know this when there are other pieces on the board. How does the King get to the other side? Take the opposition, and then give it away at the proper time to run toward the other side.
How to begin? The King has 3 legal moves. Kb2 may seem the most logical, but it falls to 1. . . Kb8! Kb1 fails to Kb7. Ka2 is the correct choice, for obvious reasons - it takes the opposition right away.
Hope this was fun and helpful. All you have to do is run sideways and keep a file in between your King and your opponent's. I've found this really useful in my endgames ever since I learned it. Good luck!