There is a total of 32 books you must Completely read in order to get this trophy. You can buy most of the books at the the book store in South Shopping District (The first store next to the gas station) Almost all of the books have a date you need to wait for until they are available.
Here is a Link to a list of all the books:
http://www.gamefaqs.com/vita/641695-persona-4-golden/faqs/65...You can speed up the progress by reading the Hyperspeed Reading book available 8/4. It allows you to read 2 chapters each time you read a book.