Forget the situps and crunches, they're not really that superior for ab strengthening anyway.
Straight up number one: planks, and any/all variations thereof. I pinned this ages ago when my main goal was to increase the length of time I could hold a plank... ignore the crunches and sit-ups and just worry about the planks, although the push-ups will help.
Planks are easier to do from your hands and harder from your elbows, so if elbows are too hard, start up on your hands. Just keep your body perfectly straight... no high butt and don't drop your hips down either.
Side-planks too.
Hip bridge. You can do this with weight on your lap too to make it harder.
Leg rotation. Lie on your back, lift the legs up. Either straight or knees bent at 90 (straight will be harder.) Make sure your lower back stays on the floor, rotate both legs to one side. Rotate them slowly down as far as you can without your lower back pulling off the floor and/or feeling any pain. For a bonus, take the legs not QUITE all the way to the floor and hold them there, just off the floor. Come back to center, then rotate down to the other side.
Plate twists. Start with low weight (or no weight), back fairly straight, legs low to the ground. And as you get stronger, lift your legs off the ground more, lean your upper body back further, and increase weight.
Deadlifts. Great for core and lower back both.
I'm clearly biased because I love them, but burpees are a great cardio burst core (and more) strengthening exercise. Plyometrics too, tuck jumps or box jumps if you're feeling ballsy, side-to-side jumps, jump lunges. Especially if you're wanting core strength for some kind of agility or endurance work.