I've read The Great Gatsby but didn't enjoy it at all. I'm not trying to discourage you from reading it, not at all. I have read a quite number of classics. The ones I really enjoyed and would recommend are A Tale of Two Cities, War and Peace, Little Women, Black Beauty, The Adventures of Tom Sawyer and Huckleberry Finn, To Kill a Mockingbird and all of Jane Austen's novels particularly Pride and Prejudice.
Shakespeare's plays are also considered classics. I really enjoyed reading Hamlet, Macbeth, Twelfth Night and The Taming of the Shrew. Romeo and Juliet not so much, I disliked the ending.
I'm definitely interested and willing to read more. I have to read The Count of Monte Cristo, Great Expectations, Crime and Punishment, Gone with the Wind and Anna Karenina.
I think everyone's opinion is different on what makes a book a classic. To me, a classic is a book that has survived many years and books that I can relate to in terms of themes. I think we can consider books that will never lose popularity as classics. Also, I regard morality to be one of the most important characteristics. If you've read plenty of classics, you'll notice that the writing style and language is very much different, kind of poetic.