Many experts believe that Shakespeare's first play was Henry VI, Part I, between 1589 and 1590. He completed Henry VI, Part II and Part IIIfrom 1590 to 1591. There is always some uncertainty about the dates of Shakespeare's plays, and no one seems to give definitive dates for all the plays. I have provide a link for a time-line, which might be of interest, and a link to a second web site that concurs with that date. On the third web site, only a date for the first performance of the play is given: March 3, 1592. Ppreliminary comments on the third site list the historical documents relied upon to arrive at the dates given on the website, while the first two web sites do not state how dates were arrived at. This suggests that there are no documents that clearly establish a date for the writing of the play, only for its performance. Since documents do not exist to show that any other play was performed any earlier, I would have to say this was the first.
It is rumoured that William Shakespeares first play written was the 3 parts to Henry IV.