Both are correct, but:
1) The first situation sounds broken, because you are informing a person about what he likes, which is strange by the principle. You never tell anyone what they like, unless they've suffered an amnesia and you are replaying all their life back to them. Who does that?
2) There is a small difference between them:
Person X likes doing Y
Means that the person does enjoy the thing in question, they take the pleasure from doing it.
Person X likes to do Y
Means that the person likes to do it for the feeling it creates, kind of. Like losing weight. You can say:
Jane likes to lose weight
Meaning she is happy she decided to do so and is doing it.
But if you say:
Janes likes losing weight
You are describing how she feels during that situation at that very moment.
Came up with another, more obscure example, just to prove a point. Sorry for the awful sentences.
He likes killing people.
Means that he enjoys doing it, at that very moment.
He likes to kill people.
Means that he's generally happy he's a killer and he likes to find victims and kill them.