Ah, such a difficult question! As a teacher, when I teach a Shakespearean play, I rediscover elements about a play, and it becomes quite beloved. I quite enjoy Macbeth, and it is an excellent play. However, I do have two favorites, a comedy and a tragedy.
A Midsummer Night's Dream never fails to make me chuckle, smile, and just be enchanted by the play. I am truly transported to another time and dimension, where lovers struggle and come together, and there is another world of fairies interfering with our world. Plus, the scenes with Bottom as a Donkey are hysterical! I love that play.
My favorite tragedy has to be King Lear. Father-daughter relationships, loyalty, dementia, manipulation, and a sad, old king who is, as he loses his power, loses his humanity. I have found the scene between the joker and King Lear the most poignant in the entire play. It fills me with an incredible sadness, which stays with me for the rest of it. My training should have me pick up a copy of the play and just demonstrate why it is poignant, but I am just going to leave with this thought: your question about why Shakespeare's works have transcended time was answered before it even began: because they speak most deeply to our hearts.