The goal of staking is to maintain the typical fountain shape of the peony bush but keep the flowers from falling in the mud. Many peonies will be self supporting and not need staking. Location makes a difference to staking requirements. A plant grown in good soil with adequate sun and moisture will develop stronger stems. Weak stems cannot support heavy heads.
When selecting peonies choose flowers that are described as single, Japanese, anemone or semi double. Avoid any that have multiple flowers per stem or side buds. Festiva Maxima, Mons Jules Elie & Mary Eddy Jones are very tall with weak legs and multiple blooms per stems but they are also very pretty and readily available.
When staking is necessary be careful with any kind of wire or cord as stems frequently break against the wire/cord. Try to avoid wrapping them up and making them look like they are in a corset.
There are commercial cages/stakes available but most are quite expensive. Some people use tomato cages but run the risk of injuring the peony roots. The Brits use twigs and branches from just pruned shrubs.
Consider the following alternatives to staking. Peonies grow toward the sun so need support mainly on one side – a daylily or Autumn Joy sedum planted in front of peony is often enough to provide support it needs. Grow them behind (north side of) a split rail fence in a country garden. I have Mons Jules Elie planted on the north side of a deck; he lays his heavy heads gracefully on the deck floor.
The following doubles do not require staking in my garden: Better Times, Cincinnati, Dandy Dan, Dragon’s Nest, Highlight, Karl Rosenfeld.