fuzzy you shouldnt ask me gym questions! i can talk for britain anyway, but when its my favourite topic........im off on one! 
you asked for it!
there are 4 muscles that you need to work to get a good core:
1, rectus abdominis
2, external and internal obliques
3, transversus abdominis
4, erector spinae
the exercises for 1 are curl movements- bringing shoulders approx 30 degrees from the floor and reverse curl movements- raising the hips and lower back from the floor, by moving bent knees towards your body.
exercises for 2, are side bends. sit on the floor with a medicine ball (pick a weight) and raise your legs up off the ground a few inches and then turn side to side and touch the floor on each side with the ball. also side planks.
exercises for 3, are planks and pulling in you stomach as far as you can and holding it.
exercises for number 4 are, back raises- lay face down on a mat, hands behind head and lift your torso and head up. superman exercises- lift opposite arm and leg up.
i would ask a gym instructor to show you correct techniques, so you dont do more harm than good. but i must warn you, that the majority of them are downright rubbish. see if you can spot a good one.