Here is a scene that illustrates how to hear the differences between two accents: HE LIVES ON SESAME STREET, DUMBASS For more of that, here's this post that was floating around Tumblr.

For media, I'd honestly recommend bypassing TV and movies and going to documentaries. The best example I can think of right now is this episode of This American Life. I haven't started listening yet, but the SFA podcasts are likely to have good accents. And then there are the videos by the Southern Documentary Project.

What is selling these days is redneck everything, so you probably want to concentrate on rhotic accents -- not the Savannah accent that Ed Helms does (he and Brian Baumgartner are both from Atlanta, which makes that scene delightful for me), but something closer to what Jenna Fischer is doing, where the speaker leans into their R's rather than elide over them.