Meh, in most cases a $500 bike will buy you $500 worth of bike, regardless of the brand. Common brands you might look at are Giant, Merida, Avanti, Specialized, Scott, Trek, Kona, Norco, Garry Fisher, GT and others than slip my mind right now. All of those brands will have something entry level around $500. Make sure you go to a bike shop to buy it too, not a department store or anything like that. Also make sure you get a test ride to make sure it's the correct size for you.
While you can get a better deal second hand or if you build it up yourself for a newbie to bikes you are likely to miss something and it will end of not being worth it as it may not run well or you could get ripped off.
Again make sure you test ride it. If you are riding on loose gravel then a mountain bike is probably a good way to go. For your height you might find the 26" wheels better than the 29" which has become more and more common for mountain bikes.
If you have a limited budget then make sure you factor in other gear you will need. Helmet is a must as well as simple gear to change a flat tyre. Also consider if you will be riding at night etc. for lights and other equipment. Don't be afraid to ask the bike shop for a deal if you buy the bike and a few other items from them as a package.
Bikes and gear can get expensive but if you are using it regularly then it is worth it.