The phrase is 책을 읽는 친구 meaning 'friend who reads a book'. In this case, you are saying 'I like the friend who reads a book. So you add 나는 at the beginning then the phrase 책을 읽는 친구 then 를 to make it the object of 좋아하다. I always try to pick out the phrase, in this case 책을 읽는 친구, And then go from there.
Honestly all of this stuff confuses me too. But my korean friend tells me that you would say 나는 친구가 읽고 있는 책을 좋아해요. So you would use both 는 and 가 apparently. (I also had to ask him why it was 읽고 있는... Its present progressive tense apparently. In case you have the same question.)