I just love peonies and have an herbaceous peony and a tree peony.* Herbaceous peonies have green stems and will go dormant in winter and disappear.* Tree peonies also go dormant, but they are more like a shrub and their woody stems remain visible above ground.* Do you know which you have?* This site explains how to propagate peonies.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.