Third Person (Limited Omniscient), then First Person
The narrative voice of Portrait of the Artist is one of its most spectacular features. Joyce was a pioneer of the stream of consciousness technique, which is a style of writing in which the narrator relates everything that happens in the main character’s mind as it occurs. We see this most clearly in Chapter One, where we’re basically inside young Stephen’s head, and we go with him from moment to moment. In the following chapters, the narrative voice is still intimately connected to Stephen’s thoughts and memories, but it skips around in time a little more, sometimes even skipping years over a paragraph break. Throughout the book, though, the important thing to note is the proximity of the narrator to Stephen – this is a majorly limited "omniscient" narrator. We never get to see inside other characters’ heads; instead, we see them the way Stephen does. The voice knows what Stephen’s thinking and feeling, but it isn’t identifiable as Stephen.
That is, until the Great Narrative Shift of Chapter Five. All of a sudden, we actually do get a glimpse of Stephen as related by Stephen. The final section of the book, which is composed of Stephen’s diary entries, is narrated in the first person by you-know-who. This is super important; through this shift in narration, we see Stephen finally stepping up to take control of his life (and his story) after his decision to leave home.