What order should I progress in Hollow Knight?

The game should guide you. If there is a place you cannot go, it probably means you haven't unlocked the ability for it and should come back to it later.

As Hollow Knight is a metroidvania, there isn't a quest indicator as such. You progress by exploring and discovering more things.

With that said, for the most part, if you can get fairly deep into an area, you can explore most of it without any additional ability. Whether you survive is just a matter of skill.

If you're talking about progression in terms of power, you unlock abilities that pretty much lets you get into more areas, charms that augment your character, charm slots (for more and powerful charms) and max. health/soul capacity. You can find these by wandering around and following the game's flow.

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The route I took was:

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If you just want to blindly follow a guide, the Wikia has a list of walkthroughs you can follow.