How does one dig up a old peony and move/divide to a new location?
Unless you absolutely have to move it I'd leave it until fall when they go dormant for the year. If you need to move it now I wouldn't divide it, just move as much rootball with surrounding dirt as possible. If you can possibly wait until fall you'd have a better chance of the plant surviving.
In fall, just dig around the entire plant, start digging 10-12 inches out from the where the outermost stems are growing from the ground. Once you've gotten all 4 sides started just rock the plant a bit, you'll be able to tell if you're deep enough based on how easily you can sway the entire plant. Depending on your specific variety, a shovel deep might be enough but you might have to go down 18-24 inches. Once the entire plant shifts pretty easily just lift out the entire thing. Knock most of the dirt off the rootball and you should be left with a brown, tuberous mass with lots of small eyes that look like those you'd see on an old potato. Those eyes are where your new growth will come from next spring. Try to identify an area where the ball isn't really dense and away from clumps of eyes, that's where you'll want to cut. You can use a knife or the shovel to break the rootball into several pieces each with several eyes. You typically don't have to be very gentle, peonies are fairly hardy to being divided this way. If pieces break off with no eyes they still could grow but will probably take several years to flower so plant those, too. Make sure you pick a new spot that gets mostly sun and that isn't boggy, peonies don't like a lot of shade or really wet conditions, well draining soil/areas are best.
If your peonies are flourishing that's a good indication you have them at a good depth for your area, just re-plant your divided peonies at about the same depth and they should be fine. If you've had fungus issues you'll want to treat the rootballs before planting, I've never had problems just putting them right back in the ground in their new locations. You can work in compost or your preferred fertilizer in to your planting hole. Make sure you dig a big enough hole that the soil is loosened several inches around the entire ball and that there are no air pockets around the rootball once you backfill and that the eyes are planted facing up no more than 2 inches deep or at the depth that works for your area. You can cut off the dead/dying foliage a couple of inches above the ground, too. Most herbaceous and Itoh peonies are pretty forgiving of being divided and will come right back up in the spring. If you have tree peonies to be divided please start another string and ask specifically about dividing/transplanting them, I've never divided one and there are different techniques. Dividing is kind of nerve racking the first time but you'll find that they respond very well as long as you're dividing in the fall. Good luck and keep posting back!
I agree with Elizibeth and only will add, that you will find the roots very brittle and will break when handled. This breaking can be controlled if you wait a day or two after digging and let the roots relax and become limber before you do the dividing. Al