In 1592, Shakespeare was criticized as being an "upstart crow." This was said about him in print by a rival dramatist. Thus, we know he was writing plays by then.
In between 1592 and 1594, the plague closed theaters so we don't see the evidence between those years. But when they opened back up in 94 Shakespeare was a part of starting up the Lord Chamberlain's Men, an acting group. Some evidence of Titus Andronicus, The Comedy of Errors, and Henry VI being written and performed exists.
It is not known for sure exactly when Shakespeare started to write his own plays. It is not even known when exactly his plays were first performed. It was too long ago and the records were not sufficient.
Here's what is more or less known. Shakespeare was born in 1564. By 1592, there is some evidence that plays of his were being performed. By 1594, there are publicatiosn of some Shakespeare plays. Shakespeare wrote most of the works that we know about between 1598 and 1613. He died in 1616.
It is very difficult to say when was Shakespeare actually started his career as a playwright. His earlier plays are believed to be Titus Andronicus, Henry iv, Richard iii, The Comdy of Errors, and The Taming of the Shrew.
It is assumed that, through wriitng the tragic play Titus Andronicus in 1590, Shakespeare appeared as a playwright.