Short answer, this is unknown.

Longer answer, so is almost everything about Shakespeare's life. Day to day records weren't kept for most people (much less, surviving ones), and without a diary or similar personal journal, little can be known about the life of a commoner like him. Stephen Greenblatt's book "Will in the World" is the best reconstruction of Shakespeare's life I am aware of, and is based more on analysis and reasonable conjecture than on recorded documentation. For what it's worth, I don't recall it mentioning Shakespeare travelling abroad. Shakespeare was in London from his late teens, and would not have had the means to afford extensive travels until later in life, when his presence would have been sorely missed by the troupe with whom he worked, and the family he supported.

Many of Shakespeare's works were based on or inspired by already extent works - Hamlet (Denmark) can be identified as a re-write of a play usually called "Ur-Hamlet" which was known a decade or more before Shakespeare's version (and is published in the back of the Arden Shakespeare edition of Hamlet, if you're interested). Many of his Italian settings, such as A Midsummer Night's Dream, are derived from classical references (Theseus is the mythical founder of Athens, Hippolyta features in the legends of Hercules, Titania and Oberon are names going back centuries, etc). Shakespeare was an expert at taking things that were already in the public mind in some way, and creating enjoyable stories about them that people would pay to see. He was a commercial playwright first and foremost - though well-regarded and successful in life, the "cult of the bard" as it's sometimes known was a development well after his death.

From looking at the context and wordplay of Henry V, one can at least surmise that Shakespeare was either fluent in, or worked closely with somebody who was fluent in, French. It is no small thing to write a complete scene (H5 III-4) in another language, based entirely on puns between the two languages. French would not, however, have been an unusual language to possess, and does not evince anything related to travel, either way.

Follow up; So we can't know about Shakespeare, but what of other middle class types of the era? I know the 'Grand Tour' was popular among the wealthy, and pilgrims have traveled since before Chaucer, do we know of any regular types who took trips for pleasure?