From my experience, it is best to plant spring planted peonies in the cool shade so they are not too stressed out by sun and heat. In the fall, move them to their permanent location. I did this last year, the peonies that were grown in the shade are now awarding me with buds but the other ones are stlll struggling.

Great thank you! Does it matter which way I place them in the ground? It looks like a lot of the growth is growing the wrong way compared to how the root itself looks, if that makes any sense. For example, the #1 root, should I plant it the way it's pictured or turn it so more root is underground? If I turn it, the growth will then be sideways, will it find it's way upright?

I think you should plant them in a way that most of the eyes are faced upward. The roots can be placed sideways or at an angle, but NOT upside down. I am sorry to say but most of these shoots will not make it. They will most likely rot. If you see these growths wilt, don't panic. They will be replaced with new growth. Just leave them alone and let them grow feeder roots and make new growths. There maybe very little or no activity for quite a while. Don't disturb them and they will send up new growths when they are ready. . Also, plant them no deeper than 2 inches and plant them as soon as possible. In the fall, move them to permanent location, add some bonemeal and plant them 1 inch or less deep. Then placed about 1 inch of mulch over them. When they start growing next year, remove the mulch off the young shoots and hopefully you will have some nice flowers.

I forgot to add that when you move the peony to the permanent location, make sure it is one with full sun as the more sun the peonies get, the more flowers you will have and the less fungal disease they get.