Peonies are perennials that do great when planted in the fall.
If you purchase peonies via mail-order companies or online, most will ship you bare roots. Each will have one or more “eyes” on it, which should be planted facing up and covered with an inch or two of soil.
Water it in and wait for spring when new growth should emerge fairly early. You should get a bloom or two that first year.
If your peonies are getting overcrowded, or you’d just like to share them with others, you can also dig them up in the fall. I usually do this after cutting back the foliage. I dig up the roots, separate them into pieces with three to six eyes on each, then share and replant.