On poetry and verse

We've been delighting in verse since Harriet was born. We've read Old Possum's Book of Practical Cats, When We Were Very Young, Jelly Belly and Alligator Pie by Dennis Lee, and we've just started Till All the Stars Have Fallen: Canadian Poems for Children. Reading the poems aloud has been tremendous fun, and Harriet likes to listen (most of the time), and I've enjoyed rediscovering poems I read years ago and reading many others for the first time-- I'd never read the Eliot or A.A. Milne before. I do wonder, however, how much the fun we're having with verse constitutes anything to do with poetry proper. Will the one lead to the other for Harriet, and for ourselves? And what's the relationship between the two? Is verse the pop music of literature? How does one cultivate an appreciation for poetry in a child? By which I mean, what is the way from Macavity the Mystery Cat to Prufrock?