Re: Getting rid of old black plastic?
I would just rip it out wherever you find it. We had that kind of property, and I always carried a sharp pocketknife for just such an emergency. If you can, find the edge and peel back. If it is too invested around shrubs, just shred out what you can.
In terms of helping the soil recover, it partly depends on how long the plastic has been there. 5 yrs? 25 yrs? Your profile says you are in IN. Unless you are planning on putting in a lot of plants in a particular area soon, it shouldn’t be too hard.
Rip out the plastic, loosen up the packed soil surface some with a garden fork or other tool, add even an inch (two is better) of compost, or manure and topsoil. Or good soil from a different part of the yard. Or put down the compost and then a thin layer of good soil, to get the soil organisms into the compost. Give the area one good watering. Odds are, the soil will recoup over a fairly short time in your area.
If you are looking at intensively planting a particular patch, you may want to do more in amending the soil for that patch.