There is no reason to believe that Shakspere did not travel. He may have been in the army for a period and he could have been a tutor. It is not inconceivable to believe that he did, over the course of his life, come to know enough of all of these subjects to create the material present in his plays.
Also, Shakespeare was most likely educated at Stratford Grammar School. He would have been exposed to a curriculum of Greek and Latin and could have featured foreign languages. The actual Latin book that may have been used in the school’s syllabus was actually mentioned in the Taming of the Shrew.
Through his work with a royal acting company, Shakspere would also have likely been exposed to the workings of the aristocracy and courts, though there is no evidence for this.
Also, many of the best writers of Elizabethan England did not come from noble stock. Ben Jonson was the son of a bricklayer. Edmund Spenser's father was a tailor. Christopher Marlowe was the son of a butcher. The idea that only someone with access to a noble education could produce great literary fiction is utterly ridiculous.