What do you think of when you think of "working your core?" Doing a gazillion crunches?
In running, developing your core is more about stabilizing your pelvis... which will make you a more efficient runner, less prone to overuse injuries. What are your pelvic stabilizing muscles? The deep abdominals (transverse abdominus, internal obliques) and the hip stabilizers (hip abductors, external rotators). It's more about developing proper "neuromuscular communication" between your brain and these muscles than strengthening them.
First, you want to locate your deep abs...
Stand tall but relaxed and reach both arms straight up over your head. Pull your navel in toward your spine as you try to reach up higher. You are now contracting your deep abs. Hold that position for a count of 10 then relax. Do it again and then again five or so more times. You can do this little exercise whenever you think of it. Now, when you go out for your run, think about doing this... run tall, engage your deep abs--pull your navel in toward your spine.
An exercise we all know--the clam, for your hip abductors and external rotators. Lie on your side, both legs bent 90 degrees, knees together, bottom arm either bent at elbow and up under your head (as shown below) or head propped up with your hand, arm bent. Now slowly rotate your top leg up (knee stays bent and foot stays in contact with bottom foot), and then slowly lower it back down. Do another 10-15 times, then do the same on the other side. Do this exercise two or three times a week.
You can also keep your legs straight, which will work the muscles slightly differently. Again, lie on your side, with legs straight, feet together, head propped or arm up under your head. Slowly lift top leg straight up, foot dorsiflexed, as high as you can, and then bring it back down slowly. 10-15 reps and repeat on other side.