About 45 minutes East of Idaho Falls is the Kelly Canyon Ski Resort. If you follow the road past the ski resort you will enter Forest Service and and will find lots of dirt roads, four-wheeler trails, and game trails that are great for running on. As an added bonus, during August and September there are quite a few places where you can find Huckleberries growing.
There's plenty of elevation gain and lots of trails, both on the side of the Canyon where the ski resort is, and also on the other side where Buckskin Morgan Ridge is. During the winter months, you will have to pay to park at the Ski Resort even if you just plan to explore on foot (or on snowshoe) in the area. I think its around $5.00 for a day parking pass. There are cross country ski and snowshoe trails in the area maintained by the ski resort.
During the summer there are places to park off the side of the road in many areas. Especially during Huckleberry season the road can be quite busy. I've also seen some Moose wandering around in the area, so you do need to be aware of your surroundings.