[gardenmermaid]

There are both heirloom and hybrid pepper varieties, also open pollinated (OP), which means you can save seeds from it and it should grow the same type of pepper. All heirlooms are open pollinated but there are also OP varieties that were developed too recently to be considered heirloom.
More important than your zone is how many days you have in your growing season ( between the last frost in spring and the first frost of fall) and how hot and humid/dry your summers are.
My growing season is only 93 days, very dry, and the weather swings from freezing to 90 degrees in about a week. So peppers can be a challenge. I have the best results from growing them in self watering containers and I do best with peppers that mature quickly.
I have consistently great results with:
Cayennes
Turkish cayennes
Jalepenos
Tam jalepenos

For sweet peppers:
Jimmy nardellos
Sweet banana peppers
Albino bullnose
Iko Iko
Hungarian red
Purple bell
Sweet cherry red
Sweet chocolate
Corno di torro Rosso ( the yellow version never does as well as the red for me)

I also have had success with growing peppers from seed saved from some of those little sweet snacking peppers from the grocery store. High mowing has picnic peppers that look like these. I might give theirs a try this year.

Hope this helps