We never get to learn the identity of the speaker of this poem – but since Sylvia Plath was a confessional poet, chances are that this speaker has quite a bit in common with Plath herself. (We know, we know, you're never supposed to assume that the author of a text is also the character that we meet in the text – but in this case, the line between the two is a bit thinner than most.)
"I" is pretty ambivalent about participating in the events of this poem – even though the 'event' happens to be the birth of her child. If you're paying close attention, you'll notice that "I" doesn't even show up until halfway through the poem (line 7, to be specific).
Even though "I" is the mother of "you," we're not about to see her spouting any of those hysterical, cutesy little love poems that we traditionally think of when we think of new babies. In fact, the speaker is not all that sure how to approach this child. At one point, she even disavows her privileged role as this new being's mother.
By the time the poem resolves, however, You and I seem to have formed a sort of bond. I marvels at the way that the new baby blows sounds into the air, seeming to make a song of her own. Even if the speaker is not all that thrilled with the new mother-role that she finds herself in, she gets around to admitting that the baby itself is pretty amazing.
Well, those are our thoughts on the speaker, but let's talk about the baby, "you" too, just for the heck of it.
It's evident early on that "you" is a newborn. But who exactly is this newborn? More importantly, what is she or he like? Well, that's what the entire poem tries to figure out. Layering metaphor upon metaphor, the speaker crafts "you" into a multi-layered being, one who's somewhere between a statue and a cat. Confused? Well, maybe that's because newborns aren't exactly known for their expressive personality. In fact, our speaker is trying pretty darn hard to come up with some sort of language to describe a being that seems entirely indescribable.
Why address the baby as "you"? Well, for one thing, it helps Plath hone the emotions that the poem develops. See, "Morning Song" isn't a philosophical mediation on the nature of babies and mothers everywhere. Nope, it's far more personal. It's about one mother and her experience with one baby – you. And directly addressing this baby asserts the close connection between speaker and subject.