I dont know how to describe it too well, but just take a deep breath in, while focusing on abdominal breathing, and basically when your abdomen area comes in when you breath out, try and keep it remaining in.
If you can just try doing that every now and again then it would become second nature to not let it go. I've just been doing it so long I forget half the time, and if I ever try and let my tummy out and to stay out, it kinda becomes a big bother between uncomfortable to slight pain.
Edit: Exercise wise, you can't really isolate train specific muscles if you could just do a lot of cardio, and do a semi full body regime. Basic strength core exercises include squats, deadlifts, bench press and overhead press.
But if you specifically would be curious on more core training, I would suggest cardio within yoga or pilates, if you want to go the more traditional gym style, you would want to do crunches, oblique (side) raises, planks, squats.. You could also incorporate good core structure within push ups, leg lifts, deadlifts, etc... And remember to get in cardio, so it will maintain more of a lean/toned muscle rather than bulky muscle tone. I would recommend cardio before strength training, and then a lesser but not quite cool down cardio afterwards. As a motto within heavy lifters would never recommend cardio after a workout session, especially when they want those gains.
Kate <3