Spiderwort is a native wildflower, and doesn’t really need any care. You can fence off your little patch to keep the dogs off if you like. It can be divided right after blooming, or in the fall when it goes dormant. That way you can spread it around a bit. Deer do eat it (at least they do in my yard!), so a little Liquid Fence is helpful in the spring to train the deer away from it. (might keep the dogs off too).
Good source for info on wildflowers: Southeastern Wildflowers by Jan W. Midgley. Gives info on identifying, propagating, etc.