go swimming and go to swim

I had this lesson about saying I go swimming and I play golf in the book, New interchange Intro.
Suddenly one of students asked about the difference between go swimming and go to swim.
I've learned that go to swim is wrong. Every grammar book says, go swimming. But I don't think that's wrong.
What's the difference?
I went to swim today.
I went swimming today.

Please help me.

I think both are acceptable, but normally used with a different meaning.

Went to swim: you went to a lake, river, sea, etc. specifically with the purpose of swimming rather than rowing, fishing, etc. (emphasis on the purpose).

Went swimming: this implies that you went to a swimming pool or a beach, that is, a place where you normally go for swimming at the first place.

However, "went swimming" seems to be much more common than "went to swim".

Thank you so much.
Here's one more.

He was absent today.
He absented from school today.

why does this word, 'absent' can be both used as adjective and verb?
I used to say if I wasn't attend a class, I was absent, which is very common,
and what I think is,
I absented myself from the class is more formal way to express the situation.

Am I correct?

You are somewhat correct. The verb use of absent is much less common than the adjective usage, but using it as an adjective is by no means informal.