If you DON'T want to replay levels, the order of acquisition for each weapon is as follows:

1. Red Revolver - Prologue
   
2. Red Shotgun - Prologue
   
3. Red Machine Gun - City
   
4. Blue Shotgun - Machinery
   
5. Blue Machine Gun - Machinery
   
6. Red Artillery - Clockwork
   
7. Blue Revolver - Clockwork
   
8. Blue Artillery - Caverns
   
9. Golden Shotgun - Caverns
   
10. Golden Revolvers - Tower

Be sure to check every section of this guide first if you're intending to get everything chronologically.

First off is the Red Revolver. It's the first one you get and must be taken to advance.

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The Red Shotgun is further up ahead in the first level (Prologue). Unmissable as well.

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The Red Machine Gun can be found while going through the City level. It's found at the end of the long hallway you enter after first fighting the cowboy enemies.

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The Red Artillery can be found after the first room with enemies at the beginning of the level Clockwork, hard to miss like the rest.

On the level Machinery, where these swinging axes are, if you jump down you can find a walkway which leads to the Blue Shotgun.

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On that same level, right before the mini-boss, you can find the Blue Machine Gun.

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On the next level (Clockwork), you can find the Blue Revolver in this big room, up where the next image is looking towards. If you've acquired the Blue Shotgun beforehand, getting up here should be trivial at best because of it's right click. (This location has been made more obvious as of 0.7.20)

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On the level Caverns, if you go to the upper level which is where the next image is looking, go through the arches then stick right to the wall and you'll find the Blue Artillery.

I made a video on the locations of the golden weapons, so instead of explaining where they are I'll leave it here.