Well, I suppose that one cannot visit Verona and miss Juliet's House and Balcony. It is only a few minutes from Piazza dell'Erbe, and is best visited early in the morning, before swamped with tour groups.
Follow the signs (or the masses of Let's Go backpacking teenyboopers) to the small courtyard at 27 Via Capello. There you will see a bronze statue of Juilet and the balcony supposedly immortalized by Shakespeare. On the rare times when it is empty, the courtyard has a romantic air, with leafy vines overgrowing the attractive graffiti-bedecked walls.
Visiting the courtyard is free, but there is an entrance fee to the building, which has a mediocre museum. (Fret not, you will be able to buy cheaper Romeo-and-Juliet kitsch in other souvenir shops around town.)
La Casa di Giulietta was an inn owned by the Capuletti family. It has been given its official title because Verona began to take advantage of tourism in the 19th century. Just like many places now offer "movie-set tours" for the masses, the city further capitalized on the story by arbitrarily placing Juliet's tomb (Tomba di Giulietta) in the crypt of San Francesco al Corso. Romeo's house is supposedly on the Via della Arche Scaligeri. I wouldn't be surprised if some unscrupulous hotel has a "Shakespeare slept here" plaque, though The Bard had never visited Italy and relied on the play by Luigi da Porto di Vincenza for details.
But Verona, like most Italian cities, was home to feuding merchant families and romantic youngsters, so who's to say that no tragic meeting of star-crossed lovers occurred here?