To me, "I'll go for a swim" sounds very slightly more immediate (as if you were already at the pool and about to get in, or already in the swimming pool playing around but had now decided to swim lengths instead) than "I'll go swimming", but Marie is correct that you can usually use any of the three phrases interchangeably. One other difference - "I'll go swim" is US English, the equivalent in UK English would be "I'll go and swim", but I think we (UK) would be more likely to use one of the other two options anyway.

all are fairly interchangeable, sorry we tend to be redundant, and like being able to say the same exact thing but in slightly different ways, I can not think of a different context or meaning to any one of them, pick which one you want to use at any given time

There might be some small stylistic differences but they all sound ok to me and convey the same meaning. For some reason, choices 2 and3 sound a little better. To say, „I’ll swim“ sounds like the answer to a question like, „do you want to take the boat or swim? Answer: : „I‘ll swim.“

1. I'll go swim - You probably wouldn't use this but you would be understood. 2. and 3. are the same and both are correct.