Who is the protagonist in Six Characters in Search of an Author?

It is difficult to say who the protagonist is in this play. The Director is at the mercy of being caught between actual characters and actors on stage. He must negotiate between the two groups what the play will be. However, he never fully grasps how "real" the characters are. He is one of the main characters in the play. He has a prominent role. But he, the actors, and the stage hands seem to be more like audience members reacting to the behavior of the six characters. However, you could make an argument that the Director is the main protagonist because he is the "author" they are in search of. They need his stage and his direction in order to live their character lives. Without him, do the six characters even show up or emerge in the theater? 

On the other hand, the six characters are all protagonists. The Mother, Father, and Stepdaughter are clearly the most prominent of the six. Each of these three is a protagonist in his/her own right. And each of the three antagonizes the other two in some way. So, you could argue that they are antagonists to each other. And you could clearly argue that the Actors are the antagonists to the six characters.

The protagonist is not always necessarily the "good guy" or the "good girl." It is more broadly defined as the main character(s). You could make the argument that it is the Stepdaughter. She is intent on exposing her father's alleged sin and she wants the play to be true. Much of the play and the play-within-the-play follows her monologues. The father also wants to tell his side of the story and as spokesperson for a lot of backstory, one could argue that he is the main character. The mother is pivotal in the story because the story largely follows her life. She was married to the Father, they had a Son, and then the Father left her. His son goes to live somewhere else. The Mother takes up with the Father's secretary and these two have three children: the Stepdaughter, the Boy, and the Girl. Every bit of the drama occurs because of what happens to the mother. 

In the overall play, you could pick any of these four (Director, Stepdaughter, Mother, or Father) for the reasons stated above. In the play-within-the-play (that of the lives of the six characters), you could argue for Mother, Father, or Stepdaughter. This is one of the most original plays ever written, so it doesn't conform to normal or typical kinds of analysis. This makes it difficult to pick a single protagonist. 

Who are the protagonist and antagonist in Six Characters in Search of an Author and why?

Protagonist (main character in the battle or conflict) and antagonist (person opposing the protagonist) are terms originating from and applicable to Greek tragic drama, and much serious drama since Sophocles.  But Pirandello is using the conventions of modern drama to make a dramatic statement about a philosophical notion – the family of characters in the play cannot define themselves or act “in character” because they have been disengaged from their putative creator, the inventor and life-giver called the author.  As a metaphor for the existential situation of existing without essence or purpose, the play moves into the philosophy by reversing the “normative” expectations not only of an audience, but of society itself.  “Where is our direction, our climax, our reason for being?”  This is an early work compared to the French Existential or Absurdist movements of Sartre (his seminal treatise, Being and Nothingness, is dated 1943) and Camus (his play, The Flies, was performed that same year and was how the two met).  Pirandello’s dramatic inquiry, while early, postdates such works as Buchner’s Woyzeck (1836) and even  Calderon’s Life is a Dream (1635).

        (The internal plot, it could be argued, has the father as the protagonist and the director of the rehearsal as the antagonist, in that he obstructs the father in his pursuit of getting his story told.  But this approach to the play's "plot" is unrewarding.)