Core training is one of the areas where it pays most to think in terms of anatomy, rather than just copying what everyone else in the gym is doing. In fact, while you're following my plan, don't be surprised if someone comes up to you and says something like, "That's not how you use that thing."
Case in point: the cable pully. While everyone else is using it for their chest and arms, you're going to camp out for 30 grueling seconds in an isometric Pallof press hold. You'll also forgo the conventional deadlift for one more week and keep doing the one-handed suitcase deadlift, one of the greatest movements there is to train your lateral stabilizers, particularly the quadratus lumborum and the internal and external obliques.
And see that $8 plastic ab roller most people think is another useless gimmick? It's actually a great ally in your quest for the abs of your dreams—at least until next week, when you meet its even more difficult older brother, the TRX fallout!