Firedrineman, here is my two cents:
From my experience, running is a great way to lose fat. Lots of bodybuilders will tell you that running will burn up all your muscle, that is not true. Most of these guys telling you this are perma-bulkers and have never been in shape. Yes if you run marathons then of course your body will be stripped of muscle, but running a half hour about 5 days a week you should be fine. You can do intervals or just run at a steady pace. I used to run about 2.5 miles on my school track in about 25-30 min time, and I was still able to put on muscle and burn 40lbs of excess fat.
As far as the core, well that is just a fancy term for abs used by personal trainiers. Just use whatever ab machines and exercises are available and get real lean and that will bring out your abs and strengthen them.