I used to hurt my back a lot too--often due to poor posture and wrong body mechanics during exercise and other activities. Romanian deadlifts, bridge and plank and crunches are to me the best core strengthening exercises that also strengthen lower back muscles. There are a lot of free resources you can find online about these types of exercises. But if you want more structure, you may want to try FitYummyMummy
. I personally use the program and I am finding it very helpful. Btw, it's tailored for Moms like you and me.