Probably want to stay off the e file. That will create another move before you can queen your passed pawn.
At first glance, I saw nothing wrong with either Kf7 or Kg7.
1. ... Kd5
2. Kf6
You're now in the same spot. But...that gets very tricky. The position is better to reply Kf7 after Kd5. So the initial move appears to be for zugzwang.
That said, I'm weak on my pawn endgames, and I don't know an overriding principal that would help. I just played out the lines, so I'm interested in better advice than I can offer.
1. I've been practicing my endgames via the following tool
https://chess-endgame-trainer.web.app/position/1/3/0
In the position linked, the only move for white to win is Kg7. Why is this? It seems very counterintuitive to move the king away from the action. Is this just a matter of calculation or is there some principle that aligns with winning these endgames.
2. More broadly, is working with this endgame trainer a good way to study endgames for me? I am about 1500-1600 rapid but feel nervous in the practical endgames that arise for me in game and feel like I am losing a lot of half points especially under time pressure.