Some of my herbaceous peonies are more than a foot high, others only a few inches high, and one has just poked its head through the soil. I usually see what looks like a pointed maroon sprout, Then when it's about 1-2 inches high, it sprouts what looks like a little tree.
Depending on the variety, and if your ground is pretty shaded and cold, it may come up later. It also depends on how many eyes were on the plant and how deep you planted it. All of the peonies I put in as bare roots a few years ago are just coming into maturity now. They were very reasonably priced and had 2 eyes. I bought so many that there were singles, semi-doubles, bombs, etc. I don't remember if they came up the first year, but they did not look like peony plants the second year and didn't start to bloom till at least year 3. Even a few had buds that second year, but they failed to bloom. The bud never grew larger, and it dried up. I know now that if I want a stronger plant, it will cost more money.
So to put in in a nutshell, have faith and patience. Peonies are pretty particular. They don't like to be moved, some need 500 hours of temps 40 degrees and below, and unless they're tree peonies, they don't want much more than about an inch of so of soil above the roots. Once they are happy, you can ignore them and they'll produce.
The peonies in the attached are from Costco, and they should be bigger and better this year.