My rule for the kiddos getting library cards is that they have to read me an entire book. The little books we used with Kate are not at our local branch, but I've heard several people rave about Bob Books. I requested the second set of books be sent from the main branch to us at our local branch. And Miss Gabby loves the books. They are short and just a little challenging. She would sit for an hour and read them over with me.
Here's the little miss with the first book she read and her brand new library card on a lanyard.