Hi,
I recently planted my first ever peony about a month ago. They had blooms when I got them but since planting them, I've gotten no more blooms and the plants are getting smaller and the leaves have strange splotches on them. Also it looks like they are beginning to have a bloom but the little buds shrivel up and never bloom.
I'm attaching some photos, any help is much appreciated!

Herbaceous peonies bloom only once a year, the smallest buds that are not developing are probably sidebuds which bloom after the main flower. Your peony was planted only 1 month ago and did not develop proper roots to sustain blooming, it is stressed out which shows in those brown leaf spots.
It looks your plant is in full sun, so cover it with plastic chair and keep the soil moist but not wet. Overwatering could kill peony by making roots rot.
Do not fertilize it this year.

AlexUnder, thank you so much. I forgot to mention that about a week after planting them we had a terrible hailstorm and it beat the peonies down quite a bit so it makes sense that they are stressed out. I will shade them like you suggested. Again thanks for your help!!

Robin, the local nurseries sell peonies that are typically grown inside in spring and don't warn people that they can suffer if planted out in full sun. If your soil is still cool planting it in it's permanent spot with temporary shade for several weeks as Alex suggests is good, if the soil is warm already it could be better to leave them potted in a cool, shady spot and plant as bare root in the fall, just for future reference—not suggesting that your dig yours up. The nice thing is that peonies are pretty resilient and will rebound next spring. Not so nice is they don't put out new foliage or buds after the initial flush so the pretty potted plant you started out with will make you doubt yourself as a gardener when it doesn't stay nice and pretty!