The first example is correct as you have to use the gerund '-ing' --- It's the standard [go + ing] form ''go swimming''. "go to swim" is however possible but, there's something missing. You might have to emphasize a location to make it work so, the sentence would be ''go (to the beach) to swim''.

We use "go [verb]ing" for activities. The only alternative is "go for a swim".

go swimming

"go swimming" is correct!

"I love to swim during the summer months". Alternatively, "I love to go swimming during the summer months". Which do you prefer and why?