It can be "나는 (책) 읽는 것을 좋아하다" or "나는 (책) 읽기를 좋아한다".
The verb "to like sth" in here is a transive verb, so that it should be "~을/를 좋아하다" in Korean. (*sth = something)
"읽다" simply means "to read (sth)".
Now we have got two verbs but "읽다" should be an object with object markers "을/를" as in "-을/를 좋아하다" in the sentence.
However, in most cases an object should be or act like a noun according to grammar.^^
All we have to do is just making a verb to be a noun and it's quite simple.
We can make a verb a noun adding "-기" or "- ㄴ것" to the verb stem instead of "-다' which indicates the base form of the verb
먹다(to eat) - 먹+기/먹+는것
읽다(to read) - 읽+기/읽+는것
공부하다(to study) - 공부하+기/공부하+는것