What's the best way to make a 2nd degree burn on the palm of your hand and fingers stop hurting?

Ice! Apply cold water or ice - ice will hurt the first minute or so, but then should help a lot. If you haven't seen a doctor, get it checked - 2nd degree burns on the hand can lead to troublesome movement problems later, and they also can prescribe burn cream & pain meds. Naproxen or Ibuprofen in the meantime should also help, as long as you have no reasons to avoid those meds.
Silvadene (silver sulfadiazine) cream. Best treatment is to get a prescription for silvadene (silver sulfadiazine) cream if your doctor feels that you can heal with dressings. Use the cream twice daily. This not only helps heal the wounds quickly but also soothes the burnt area. Alongwith this taking Motrin or aleve will help. Some patients also require antibiotics by mouth. Some deep dermal burn wounds may require surgical treatment if not healing well.
Dressing. With a hand burn, as it it such a functional area, you would be wise to see a doc. If it just a superficial partial thickness burn , pain medications(narcotic), anti inflammatory medications(ibuprofen type), and an appropriate anti microbial dressing ( silvadene, (silver sulfadiazine) silvernylon, neosporin/xeroform etc) should help with pain. Also important to perform range o motion exercises.