I was introduced to stability balls by my trainer, who wanted me to improve my "core" strength.
We started with simple crunches (on the ball you can focus more on your abs and protect your back) and "walk-outs" (lay on the ball, put your hands on the floor and "walk" on your hands until your toes are on the ball).
We then added push ups to the walk outs (walk out, do 2 or 3 push ups, walk back, walk out, etc, 10 reps).
We then added "twists" to the walk outs (walk out and then move the ball as far as you can to one side by twisting your hips and feet, and then to the other side).
We also use the ball to do oblique crunches.
We also do the plank on it, and we do "roll out planks" - where you put your forearms on the ball and push it out as far as you can, while maintaining the plank position.
As I've gotten stronger, we've added weight plates to both the crunches and the oblique cruncehs. And the last rep of each set always involves holding the weight out, behind my head, for a count of at least 10.
I went from pretty flabby, to pretty strong (though it took about a year) thanks in large part to the Swiss balls -- love those balls!
While the Swiss ball is a great aid, don't forget Bosu! It's much more versatile. You can use it for legs too.