The speaker seems to be an innocent and playful child who likes riddles. In the first line of the poem, he sounds curious about "who made" the lamb, but by the second line it's clear that he knows the answer and wants to test the lamb. We don't learn that the speaker is a child until line 17, and you could argue that he isn't literally a child. By this logic, the speaker calls himself a child only in the sense that "we're all children of God."
But child or no, the speaker sounds young and, well, inexperienced, judging by the simple words and sentence structures he uses. He seems amazed that words can have more than one level of meaning, particularly the word "lamb." He's thinking, "So this word refers to an adorable animal and to Jesus Christ. That's so cool!" At the same time, you could accuse him of being naïve. In keeping with his role as representative for Innocence, Inc., he focuses on how Jesus "became a little child" like him, rather than on how Jesus became like a sacrificial lamb upon his death.
What's this kid doing hanging around with lambs anyway? This is a pastoral poem, so he's likely a shepherd who walks around with a big stick and makes sure the sheep don't get into trouble by falling into the stream (or however it is sheep get into trouble). He spends a lot of time outdoors and has a very basic education on religious matters. More than likely, he's just as irresistibly cute as the lamb, with rosy cheeks and a pint-sized shepherd's crook.