Now, when it comes to speakers in poetry, it's never a good idea to assume that the "I" of the poem is the actual poet him-herself. What if that poet is using a character to narrate the poem? Sound tricky? Well, it can be. Luckily for us, though, that's not the case here. The speaker is none other than Coleridge himself.
Throughout the poem, he doesn't seem to be fantasizing about anything (unlike in other far-out Coleridge poems like "The Rime of the Ancient Mariner" and "Kubla Khan"). Nope, he's actually discussing his real life. There's no fiction in the poem—it's pure memoir, talking about Coleridge's childhood, his life in school in London and in the country, and his hopes for his son, Hartley. If it was written today, someone might've called it "confessional" poetry (though, actually, that might apply better to other of Coleridge's "Conversation Poems," like "Dejection: An Ode.")
Coleridge draws the reader directly into his train of thought. We see it bouncing around, jumping to a new stanza as it changes its subject, focusing on something fresh. We get the sense that Coleridge is a dreamer, intensely thoughtful and intelligent, very religious, and with a possible fondness for solitude and brooding. But he's also clearly someone who loves and cares about the people around him. In this case, he concentrates on his hopes for his baby son, wishing him a good life.