How do I soothe a painful sunburn on shoulders and back?

Soaks, ibuprofen. Apply cool washcloths frequently to the burned area and take Ibuprofen or something similar for the pain and inflammation. And please learn the lesson that mr. Sun is trying to teach you: enjoy his light and warmth, but respect his power. Apply a good broad-spectrum sunscreen lotion every time you go out in the sun.
A remedy. I agree with dr. Eastern's wisdom. Aloe vera (aloe barbadensis) can reduce redness, pain, itching & swelling from a burn. It has anesthetic, anti-inflammatory & antibacterial / antifungal qualities. It can speed burn healing by stimulating growth of new skin. Non-steroidal anti-inflammatories and acetomenophen can be very helpful. Hydrate well and don't peel off any deep layers.