Stratford-upon-Avon, UK, the poet's birthplace, with the house where (it is believed) he lived as a child, and another house where his wife lived as a child. Inside Holy Trinity Church there's the Shakespeare's funerary monument. Royal Shakespeare Theatre is well worth attention too.
New York, USA, with Shakespeare-in-the-Park performances and Shakespeare Garden (in the Brookyn Botanic Garden) makes it to the third place. As a little additional score booster there's the Puck Building, named after (and decorated with statues of) a character from A Midsummer's Night Dream.
Verona, Italy. Similar story. The Basilica of San Zeno, the crypt of which is said to be the place of the marriage of Romeo and Juliet; Juliet's balcony; etc. A real fan would remember that there is also a Shakespeare comedy called The Two Gentlemen of Verona, about some guys who were, well, from Verona.
York, UK. Yes, the old York has made it to the list too, not only the New one. It is the ideal place to go for a couple in which one likes Shakespeare and another dislikes him. In Siward's Howe earl Siward, a character in Macbeth, is said to be buried. In one of the old city gates a small museum hides, dedicated to just how wrong Shakespeare was about Richard III.