Hello, charlie138. I would have no problem with either sentence. It is perfectly fine to list three verbs in the infinitive without placing to at the head of each one. It is not as though the second, more formal-sounding sentence is incorrect either. If I saw both in a Sentence Correction question, I would favor the more concise version but would turn to other considerations to make a sound judgment. (Eliminate what you know is wrong first, then test the rest.)

Good luck with your studies.

- Andrew

I completely agree with what AndrewN has said. Though both the sentences are correct, the second one is a bit wordy, so the first option is better between the two.

If both sentences are in answer choice, I prefer first "I like to eat, dance and swim".
It is consice and streight to the point.
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Hello again, charlie138. I would like to add to my earlier post that I would expect to see the infinitive marker to if its omission in the series might lead to confusion. For instance, a similar sentence might begin,

I like to eat a snack before I dance...

I would not place a comma and the word dance right after the first element in the series, but would indicate the second element by adding another to:

I like to eat a snack before I dance, to dance as fluidly as my teacher...

Finally, since the first two elements of the list begin with to, I would introduce the third in a similar (parallel) way:

I like to eat a snack before I dance, to dance as fluidly as my teacher, and to swim as gracefully as an eel.

You have to lean on contextual clues to make a determination sometimes. Again, in isolation, I would go with the more concise sentence you offered in the original post. But do not follow recommendations blindly. The clear expression of vital meaning should be your top consideration in SC.

- Andrew