As promised, here are some good exercises to start with for core work and stability balls-• Crunches- Moving off the ground and onto a ball supports your back a bit more, and gives you a better range of motion for your crunch. Experiment with moving the ball between your lower back to high back for different ranges of balance.
• Push ups- Again, experiment with having the ball only on your toes, at your knees, and at hips for a variety of balance.
• Free weights- You can lay with legs bent or sit on the ball and use free weights like a typical gym bench plus the core benefits.
For more ideas, here is a great site with several ways to use your ball, description and pictures on how to complete each exercise as well as a section describing which muscles groups are mostly used in each workout.