Five of the Bard’s plays are set in Italy, and Richard Paul Roe maintained that five others contain Italian influence in his 2011 book The Shakespeare Guide to Italy.  Thanks to Roe, who traveled up and down Italy with dog-eared copies of Shakespeare in hand, we can state with a high degree of confidence that the main author of these plays spent a fair amount of time traveling in Italy.
The Bard possessed such a vast array of details concerning a particular stand of trees, no-longer-extant waterways, a nickname for a door, etc. that taken en toto they make it almost impossible to credibly maintain that he picked them all up from speaking with travelers and perusing books. The latter is what Stratfordians have usually claimed, as there is no documentary evidence in England or other countries that Shakespere ever traveled abroad.
One should read Roe to appreciate the breath of his research into the Bard’s knowledge about Italy. I will confine myself here to a few examples, the first from Romeo and Juliet. In this tragedy, Romeo is described as walking “underneath the grove of sycamore that westward rooteth from the city’s side” (I.i.127-8). Roe found stands of sycamores, part of a grove now intersected by buildings and boulevards, still growing outside a western gate of Verona.
Scholars have criticized Shakespeare for including characters who travel by boat between land-locked Italian cities, but the Bard knew Italy far better than they. Travel by boat was safer in his day than travel by land, and a long-lost network of waterways existed that Roe uncovered through sheer diligence. In The Two Gentlemen of Verona, for example, both Valentine and Proteus journey from Verona to Milan by water. It turns out that this could be accomplished by sailing from Verona down the Adige River, through a system of canals, to the Po River, to the Adda River, to the Martesana Canal, to Milan. The Martesana’s entrance to Milan was filled in for streets in 1928, but the canal still exists outside the city walls, while the canals between the Adige and the Po exist only on old maps.
A Midsummer Night’s Dream takes place in Athens, not Italy, although it does not refer to Greeks or things Grecian. Poe happened across a small city near Mantua named Sabbioneta that was known as “Little Athens.” Sabbioneta was carefully laid out in the 16th century by Vespasiano Gonzaga Colonna, who in particular invited scholars and intellectuals to visit. Poe learned that the main gate of the city was known as the Duke’s Oak. In A Midsummer Night’s Dream, Bottom and Quince make a plan to meet at the Duke’s Oak.
Roger Prior found telling details in Othello which confirmed to him that its author had visited Bassano del Grappa, a town in northern Italy. In Shakespeare’s day, there was a fresco on the outside of a private home owned by the Dal Corno family, a fresco now sadly deteriorated and kept in a museum.  To see an image of the fresco, click here.
Prior maintained that the Bard associated this fresco with sexual desire, perhaps because “Corno” means “horn,” and because the Dal Corno’s sold salt: the house stood in “Piazzotto del Sale,” “the little square of salt.” Shakespeare regularly employed “salt” to mean “lustful.” The fresco is divided into four horizontal bands, and in one of them, a monkey stands over a large goat. A Biblical image on the fourth band, below the goat and monkey, depicts The Drunkenness of Noah. Noah is naked, on his back, with his legs spread apart, looking like a fool. In the third band is a painting of a lady holding a torch in her right hand above her head, whom Prior reasonably interpreted as the figure of “Truth.” When the shutters to the windows on either side of her were opened, they would enclose her in saloon-type “doors.”
Iago says of the lecherous Desdemona and Cassio:
Were they as prime as goats, as hot as monkeys,
As salt as wolves in pride, and fools as gross
As ignorance made drunk; but yet, I say,
If imputation, and strong circumstances
Which lead directly to the door of truth (Othello III.iii.404-8)
In addition, Prior found that the name "Otello" was popular in Bassano, "and nowhere else" (Prior, 6). One Giovanni Otello had been part-owner of an apothecary's shop on the main square in Bassano. Another apothecary on the same square was known as "the Moor" after the sign of a Moor's head which hung outside it. As Peter Farey pointed out, "Although Cinthio's Hecatommithi, the main source of Othello, concerned both 'the Moor' and 'Disdemona' (sic), it was Shakespeare who introduced the name Othello for the main protagonist." 
This is a brilliant piece of detective work on Prior’s part: a subliminal association in a play that indicates its author viewed the fresco in situ, in Italy.
 Richard Paul Roe, The Shakespeare Guide to Italy (New York: Harper, 2011). See also Grillo, Ernesto M. Shakespeare and Italy (Glasgow: Glasgow University Press, 1949).
 Roger Prior, “Shakespeare’s Visit to Italy,” Journal of Anglo-Italian Studies 9 (2008): 1-31. The fresco resides in the Museo Civico in Bassano del Grappa. It is discussed and reproduced in Jacopo Bassano c. 1510-1592, ed. Beverly Louise Brown and Paola Marini (Fort Worth, TX: Kimbell Art Museum, 1993), 530-1.
 Peter Farey, "The Bassano Fresco," http://marlowe-shakespeare.blogspot.kr/, accessed September 20, 2013. See also John Hudson, "Heretics Foundation XVIII: Goats and Monkeys," December 14, 2009, http://www.clydefitchreport.com/2009/12/heretics-foundation-xviii-goats-and-monkeys/