What kind of dirt should I use for the worms? Should I use just regular dirt from the back yard or should I use peat moss?

You actually don't really need to add dirt to your vermicompost.  You can add a cup or two of regular soil from your back yard to add some beneficial microorganisms and also some grit.  Worms have gizzards (like chickens), so they need some kind of gritty material to help them with digestion.  Finely crushed egg shells are also great for that!

I suggest against using peat moss for anything because it's not a very sustainable resource.  Instead, for bedding you can just add a lot of newspaper, or you can also use a peat moss alternative like coconut fiber/coir, which helps to absorb and retain moisture.