They are both correct. No doubt there is some subtle difference, but I am not really sure: in the first case you would probably add the occasion, for example: "I like to read a book in the evening.", but you can also say "I like reading a book in the evening."

Both forms are correct. There is a slight difference in some contexts, but in this situation the two constructions have the same meaning. What IS strange about your sentence, however, is the singular object. If you are talking about a hobby, for example, you should say either: I like reading books. or I like to read books.