One of my favorite books for teaching kids about compassion is Have You Filled a Bucket Today? by Carol McCloud. It's about sensing the feelings of those we see every day and finding a way to make them feel good about themselves. I love the message about how you can't "fill your bucket" (make yourself feel better) by stealing from others buckets (making them feel badly) and that when you fill others' buckets, you actually fill your own too.
Compassion and kindness start with those who we see everyday. This book provides a very concrete image to help kids understand that.