Yes. To win, your King needs to reach one of the critical squares e5,f5,g5. If your King goes to e3, the black King goes to e7 taking the distant opposition and if you advance further, it will take the direct opposition and block you. But if you go to g3, they can't.
Edit: of course, after 1. Kf2 Kd7 2. Kg3 Ke7 DO NOT PLAY Kf4, but Kg4.
2nd edit: To put is as simple as possible: if you have a choice, make it hard for the opponent's King to get to the same file as your King.
This position depends on taking the opposition from the enemy king. If you move to e2, you'll notice that your king sits on one corner of a rectangle of white squares (e2, c2, e8, c8) with the black king on another corner. This gives you the opposition. If you are on the same rank, file, or diagonal as the enemy and you can move to have an odd number of squares between you, you have the opposition and can drive him back. If he moves and gains an odd number of squares, then he has the opposition and can drive you back.
K+P vs, K endgames are too complicated to go into in a reply to this post, but I highly highly recommend Jeremy Silman's endgame book which makes all of this crystal clear.
Hi. I call it long T and short T to help me remember. When the pawn is in your half of the board you need to get your king to either e5 f5 or g5 in the position above. These are the key squares. I call it long T. If the pawn were in your opponents half then the key squares are the three squares directly in front of the pawn. I call it short T. I think I've got it right. Now if both kings are racing to one of the key squares. I think you get control first by going the long way round, for the far side. Once there you can take the opposition and work you king up. So it starts as a key squares problem and then becomes an opposition problem.
BTW. No need to count odd squares. Whether direct, indirect or distant, if you move your king onto the same colour squares as your opponents king, then there is an odd number between you and you take the opposition. Dont know why the text books get everyone counting.
Thanks for your help, everyone. I knew about the idea of using opposition to make my way up the board, but I didn't know about critical squares. Luckily, this was basically the next lesson in IM Anna Rudolf's endgame course. Thanks again.
There is another famous chess site that discusses key/critical squares...with exercises. 
The placement of your king and pawn depends on which rank they are.
King vs King can make some interesting positions when you get them close. Use yours to dictate the position of your opponent but shielding or pushing.
someone already mentioned the key squares e5 f5 and g5. the point here is to try to go to the one farthest from the opponent's king, which is g5. you can visualize that in 3 moves en route you should be able to get the king to g4 and then black gets the king to f6 to stop kg5 and only then does opposition with kf4 matter, but technically it is still the fight for one of those key squares, which you should win one of since now black has to give ground.
someone already mentioned the key squares e5 f5 and g5. the point here is to try to go to the one farthest from the opponent's king, which is g5. you can visualize that in 3 moves en route you should be able to get the king to g4 and then black gets the king to f6 to stop kg5 and only then does opposition with kf4 matter, but technically it is still the fight for one of those key squares, which you should win one of since now black has to give ground.
YOU ARE RIGHT
I'm trying to do learn some basic king & pawn Vs king endgames. I'm getting much better, but then I came across this position.
Thanks for your help.