what exercises can I do at home that target these major areas?
Squats, planks, pushups, pullups, and running are great ways to target those areas.
Core strengthening. It helps with back alignment. All the different Planks, bicycle crunches (sit-ups and many other crunches can be bad for your back) stand or sit up straight and hold a weight straight above you, engaging the stabilizing muscles. I've even used bottles of water packing taped together for that as the weight.
Do research into core exercises. Good luck.