Should you need to dig peonies, do so after the plants go dormant in the fall. Cut the stems low to the ground and dig the roots out of the soil, handling them with care. You can see the "eyes," which are the buds that will grow into stems come spring time. After washing away the soil, you should divide the roots into clusters containing 2-5 eyes and some of the fleshy storage roots. Then remove any damaged parts and smaller roots, leaving 4-6" stubs of the big roots.
When planting, choose a sunny location. In garden design, peonies are best planted in clusters of three or more, or used as the backdrop of a flowerbed in order to make a bigger visual impact. It's very important not to plant the crowns too deep or the plants will struggle to bloom. The eyes should never be covered by more than two inches of soil! See more details in the K-State publication Peonies in the Garden. Mulch can add to the depth, so take care not to cover the plants too much with any material.
Peonies can take up to three years to produce a full flower display after dividing, so be patient. Once your plants have established, though, you can take comfort in knowing that you will be graced with beautiful blooms for many many springs to come.
For more information, see Peonies in the Garden, a complete guide to growing peonies. Also, Peonies for Commercial Cut-Flower Production offers detailed information regarding growing peonies on a large scale.