Direct core training should be a relatively low priority for most trainees, especially those that are capable of regularly performing free standing compound movements that tax the core muscles sufficiently, such as barbell squats, deadlifts and strict overhead presses. With that in mind there are still good and bad ways to train the core muscles directly, the good ways are those that do not reduce the concept to abdominals alone but instead tax the body in a way that relates to and carries over in functional movement:
Consider the way in which the core muscles function when the body is under load, when the spine is loaded the abdominals, transverse abdominis, obliques, spinals erectors and more contract in a static fashion to keep the vertebrae in a fixed position, allowing as little flexion as possible to occur during resistance training. This is exactly what is taking place in movements like the plank, its many variants and kneeling roll outs, making them a safe and effective way to target the entire core with useful transfer for compound movements.
Opt for movements that bring the legs up toward a stationary torso, like a vertical knee raise or garhammer raise, as opposed to brining the torso up toward a stationary lower body, like a sit up or full crunch. Sit ups and crunches pull the spine into significant flexion against resistance provided by the upper body or sometimes a free weight or cable, pulling against the stretched lower back while leveraging the lumbar spine unnecessarily. This kind of movement against resistance doesn’t really occur in an athletic context, but hip flexion does, a lot. When the hip flexors contract to pull the femur up toward the torso and the lumbar spine remains extended with the pelvis in anterior tilt (butt stuck out) the abdominals contract statically, if lumbar flexion and posterior tilt (butt tucked forward under torso) are allowed to occur with each rep the abdominals are then contracting dynamically and undergoing stretch-shortening; the efficiency and pragmatic nature of this type of movement make it extremely preferable versus a sit up or similar.
Rotation of the spine while maintaining extension is another movement pattern that occurs in athletic movement and sport very frequently, it also engages and places tension upon the obliques, abdominals and spinal erectors in a dynamic and safe manner. Think Russian Twists, Standing Cable Rotations and Landmine Lever Rotations. Opt for these movements as opposed to things likes Weighted Side Bends and Twisting Sit Ups which again do not occur in functional movement.
While we don’t afford much time and focus to direct core training because the slight, targetted movements do not expend many calories at all, an issue if the focus of your training protocol is fat loss, and the core is almost always working in other larger movements that will surely be present in a more advanced training protocol for someone seeking muscle mass or strength, it is still important to adhere to these principles if direct core training is deemed necessary.