The first performance was Julius Caesar.
The original structure of the Globe Theatre stood until 29 June, 1613, when its thatched roof was set ablaze by a cannon fired in a performance of Henry VIII and the Globe burned to the ground.
So, that was the last until the Globe was rebuilt, but I don't know what the last was at that rebuilt Globe.
The first known performance to be played in the Globe Theater is Julius Caesar.
Later on, it got burned to pieces. On June 29 1613, it went up i in flames during a performance of "Henry the Eight" when a theatrical cannon, set off in the performance, misfired and ignited the wooden beams and thatching of the theater and the whole theater collapsed down and it was rebuilt the following year, 1614.
But after that, there was no more Shakespeare at that theatre. After Shakespeare's death, some of his pieces was performed at court for Christmas celebrations.