I'd probably look at something like the Kona Dew or the Specialized Crosstrail. The Dew is an example of a bike with no suspension, which is going to be more efficient (and lighter in weight) when riding on reasonably smooth ground such as pavement or a gravel jogging path. The Crosstrail will have a suspension fork, which might be nice if you are also hitting some mountain-bike trails.

Most of the major brands -- Giant, Trek, etc. -- have bikes similar to the above two. I just threw out the Dew and the Crosstrail as examples. Oh, and those two bikes have 700c wheels, which I greatly prefer for road-riding.

If you're going long distances, you could look at a cyclocross bike. That'd give you drop bars. Not sure about getting one for $500 though.