You may be a bit limited on what you'll find for $500 in a new touring bike, but you might find a used bike that would be very good.
You also need to keep in mind that you'll need other equipment, gear, clothes, etc for riding long distances, and that stuff isn't cheap.
I'm not trying to be negative, but I don't want you to get sticker shock when you start shopping around. Also, you don't have to buy everything at once...you can get the most important things first, then pick up other stuff over time.
I would recommend that you spend some time in the bike shops where you live, not to buy, but just to see what brands of touring bikes that they carry. There's nothing wrong in telling the salesperson that you are just starting to shop for a bicycle and that you have no intention of buying one today, but if they would be kind enough to show you what they have you will certainly keep them in mind when you are ready to make a purchase.
This will also give you a good idea about how friendly and helpful the people are that work at the different bike shops. You'll probably be taking your bike back there every once in a while for maintenance, so it helps if they are nice people that you trust and like.
There's several companies that make good bikes ( Trek, Cannondale, Giant, Felt, etc.), so it would be helpful to know what your bike shops have available locally.
Once you know what brands they carry, you could post that info on here and I think you'll get some good feedback.
Have fun shopping!