Kate has given you a great article to read. I agree with this article as well.
As a sports PT, I can tell you that people do not realize how much your core is actually working when you do exercises like squats, deadlifts, clean and jerks, clean and presses, and etc.
If your trainer is telling you that your core is weak, you should ask your trainer to give you some exercises to strengthen your core. After all, you are paying him/her to train you right?
However, you should keep in mind that "core stability" has lots to do with the entire chain, not just the abdominal region. This goes from scapular stabilizers to the hamstring muscles to the deep intercostal/diaphragm as well.
One way to strengthen your core stability is to try doing more exercises with single leg stance or on uneven surfaces from now on. For example, try standing on one leg and doing bicep curls, standing shoulder presses, etc. to get your core to engage more. Also, you can try standing on an uneven surface like a bosu ball when doing exercises.
Another way to increase core stability is to perform dumbbell exercises (chest presses, etc.) on a stability ball. You must be extra careful as you may lose some balance at first.
Keep in mind that if you are going to train for just core stability, you may lose the ability to gain true strength because you may not be able to perform heavy weight liftings while you are on uneven surfaces.
Here are some of the best exercises to strengthen the entire "core stability:"
- Kneeling Stability Ball Rollout
- Stability Ball Hamstring Curl
- Dumbbell Squat Swing
- Dumbbell Golfer Swing
Ask your trainer to give you a specific core stability program. Good luck!