I agree with you that the second sentence is peculiar. In your first sentence, "right away" is not needed.
Student or Learner
Are both sentences correct? What is the difference in their meanings?
1) Most animals can swim right away the first time they go into water.
2) Most animals can swim right away for the first time they go into water.
To me, it sounds as 2 means that animals can swim only once, that is, only during their first swimming.
I see. How would you rephrase it to make it natural?
That's what 1 does. 2. doesn't have a meaning.
We agree that sentence 2 is not good English, and that 1 is. I’d hesitate to called 2 grammatical though. If Mike thinks 2 has a clear meaning, I’d disagree with that. I think ‘for’ may be a typo of ‘from’ – ‘from their first time’. In fact, that would be as good as 1, perhaps better.