I really like the Poetry for Young Children. It has an appropriate biography of the writer (about 4 pages) and 25 poems. They are lovely, with pretty full color pictures. Generally I can find nice hardbacks from Amazon for prices ranging from .01 to $3.00 plus shipping. If you are in a Scholastic club and have access purchasing, I know that they have these for sale in paperback.
We have also enjoyed a book called Selected Poems for Children, and are currently working our way through Favorite Poems Old and New.
There are some poets that you can get for free from Kindle, too. Edgar Guest is a personal choice – I appreciate his choice of subjects (usually family related) and his gentle wit.
As for memorizing, I usually just find one that we like in each of our poetry studies and memorize it.