All movements come from core. Having a strong core will reduce your risk of injury, improve your posture and physical functions, and will strengthen other exercises. Core training should be prioritized based on the complexity of the different muscle groups being worked.
Muscles and Actions
The hip flexors are complex and involve many different muscles. The flexion at the hip involves: the psoas group, iliacus, rectus femoris, sartorius, tensor fasciae latae (TFL), pectineus, adductor longus, adductor brevis, and gracillis. Hip flexor training can be as simple as lifting the leg while maintaining a neutral spine. These muscles are commonly tight on individuals who sit consistently for long periods of time.
The muscles that control the action of twisting and bending to the side are also a very complex group. The action of twisting and bending also involves muscles of the spine and lower back, and should be considered at the same time. These muscles include: external obliques, internal obliques, quadratus lumborum, the erector spinae group, and many other small spinal muscles.
The retus abdominus is the superficial abdominal muscle and is used in spinal flexion, and anti-extension. The transverse abdominus is a very important muscle used to stabilize the core, and to breathe. It is also the muscle responsible for sucking your stomach in.
One of the best ways to strengthen the core is in a neutral spine position. This means your spine does not move, and resists motion rather than create motion. The abdominal muscles are primarily endurance muscles being that they are constantly being used to some degree. This means that excessively fast motions and motions that hit the maximum the range of motion can be potentially hazardous. Focus on 3 plains of motion to train your core. The sagittal, or front to back plane should consist of anti-flexion, anti-extension, and hip flexion. Exercise examples: Planks, super mans, leg lifts. The frontal plane is the side to side plane and can be trained by doing side planks, and exercises weighting one side of the body at a time such as a single arm shoulder press. The transverse or twisting plane should consist of rotational and anti-rotational exercises. Exercise examples include: the pallof press, chops, and thoracic spine mobility drills.
Sekendiz, Betül, Mutlu Cuğ, and Feza Korkusuz. "Effects of Swiss-Ball Core Strength Training on Strength, Endurance, Flexibility, and Balance in Sedentary Women." Journal of Strength and Conditioning Research 24.11 (2010): 3032-040. Web. 7 Nov. 2014.
Willardson, Jeffrey M. "Core Stability Training: Applications to Sports Conditioning Programs." The Journal of Strength and Conditioning Research 21.3 (2007): 979. Web. 7 Nov. 2014.