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We say "go for a walk, go for a run, go walking, and go running."
Do we also say "go for a ski, go for a fish, go for a sail, go for a shop" instead of "go skiing, go fishing, go sailing, go shopping"?

Generally, no.

I don't think your examples are interchangable.

I'm going for a walk (40,000 hits)
I'm going walking (this sounds weird, only 2,670 hits).

I'm going for a run (4,600 hits)
I'm going running (5,130 hits)

going for a ski (49 hits)
going skiing (8,460 hits)
going for a fish (46 hits)
going fishing (49,000 hits)
going for a shop (5 hits)
going shopping (68,500 hits)

Iain

Munsook wrote:We say "go for a walk, go for a run, go walking, and go running."
Do we also say "go for a ski, go for a fish, go for a sail, go for a shop" instead of "go skiing, go fishing, go sailing, go shopping"?

I have never heard "go for a ski" or "go for a shop." I can imagine "go for a fish" but not in the same meaning as "go fishing." (Perhaps short for "I'm going to go to the store for a fish.") But I think "go for a sail" might be used.

So the short answer is no

"Go for a walk" includes the verb "walk" turned into a noun (uncountable);
go walking makes that sufficiently clear.
But "go for a ski" means you go to pick up a skiboard (a noun, countable).
The gerund 'skiing" is derived from the noun 'ski', whereas the noun "walk" is derived from a verb.