All three sentences are correct but they can carry a ridiculous complexity of meaning.

“I’m going for something” in this context, means I am departing to do or obtain that something. I know that something that is going to happen or be happen. Whereas in other contexts “going for swim” could mean that attempting to achieve something ( “I’m going for the record”), or I am choosing an option (given a choice of desserts, “I’m going for the ice-cream”), it could instead mean that either of these alternate possibilities applies to “going for a swim”.

What is intended to mean by “I’m going to have a swim”? Are you in a hurry to go for a swim? In the specific case of swimming, this distinction is pretty meaningless, but if you replace the activity with, say, napping, it gains some relevance: “I’m going for a nap” is appropriate only if you are napping elsewhere, while “I’m going to (have|take) a nap” would also be appropriate if you are going to nap right where you

are.