Go along the path.

[bruxinha]

First, I'd like to wish all mods and users a happy and successful new year!

Now to my question:
When giving directions, students learn phrases like go straight on (or go straight ahead), go down/up the street, go along the path, etc.
Examples (from my students' dialogues):
Go straight on and take the second on the left.
Go down the street and turn right at the next corner.
Go along the path and when you see the river, go across the bridge.

Would it also be possible to say "go straight on the path..." in the third example? Does it sound natural to a native speaker?

I'm not sure about "straight on" in this sentence, since we're talking about following a path. There is no picture to it, but paths are commonly (more or less) winding and seldom straight. It may be grammatically correct but doesn't it sound a teeny bit awkward? Or would native speakers also say it naturally instead of "along"?

 

[emsr2d2]

It would be unnatural to specify the kind of walkway being referred to if one were pointing to the walkway or to a map of it. If you were trying to describe how to get somewhere without the benefit of a map or of actually being at the location, I suppose you could say "Carry on straight down the path/road/avenue/track ...".
It really doesn't matter if the walkway in question is winding or straight; we'd still use "straight on" to indicate that the person must not actually turn off that walkway.

 

[SoothingDave]

"Follow the path" is natural.