Everyone is pretty much spot on. You're Mazda isn't really equipped for the trails, it would be ruined in the first 100 feet. There are some forest roads up in the mountains that you could drive on. Look for Forest service roads, however there are some that can get rough and aren't maintained as well. I'ts been some time but, there is a road that runs along the Linville Gorge that at one time would be passable in your vehicle. I haven't been on it in years but You would probably enjoy it.
Old North Carolina 105 I think is the name.
All of this is at your own risk, and I'd advise going with someone if you are inexperienced.
Hope it doesn't discourage you, and look into it more. Maybe if you have a friend with a Jeep you could ride passenger and see what we are talking about.