We have no reason to believe that Shakespeare ever went anywhere except Stratford and London and the places in between. It is of course possible that he may have travelled during his lost years 1585-1592, but we would have no idea where.
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The only travelling we know that Shakespeare did was between his home town and London. He may have been seeking fame and fortune, but it is more likely that he was really seeking a job.
We don't know, really. By that point he had probably established his theatrical career. It wasn't a plague year. Perhaps he didn't travel at all.
Shakespeare did his work in London to keep William Shakespeare alive.
There was a plague outbreak in London, one of several during Shakespeare's career which closed the London theatres and caused the playing companies to go on tour.
During the late 1580's and early 1590's Shakespeare traveled back and forth from London.
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Eventually, he went to London. However, we do not know much about the period of Shakespeare's life between his leaving Stratford and being a successful playwright and actor in London. He may have gone somewhere else in the meantime.