A peony asks for three things from its spot: six or more hours of direct sun, soil that drains well, and about a foot and a half of open space on every side. Get those right, and the rest mostly takes care of itself.
Here are the five things I look at before I dig, and none of them are complicated.
How much sun do peonies need?
Peonies want full sun — at least six hours of direct sunlight a day, and more is better. They can survive on less, but they won't get as big and they won't give you as many blooms. Full shade is a no-go.
Soil
They like rich soil with organic matter in it. If you're working with poor soil or heavy clay, prepare the site ahead of time by mixing in plenty of compost.
Water and drainage
Peonies do not like standing water. Find a spot that doesn't hold puddles or stay muddy after a rain. If other plants struggle to grow there, that's your answer.
Space
Give your peony at least a foot and a half of clear space on all sides — more if you can. Keep it away from other peonies, from flowers, trees, and shrubs, and from the foundation of a building. It will get bigger than you expect.
Your growing zone
Most peony varieties grow in USDA zones 3–8. That covers most of the country, but not all of it, so it's worth checking your zone before you plan. You can look up your USDA growing zone by zip code.
That really is about all there is to it. I like to look out the window at different times of day and watch where the light goes before I decide anything. It's the closest thing to a secret I have. If we both follow these five steps, we'll both do great.