What kind of doctor to see for wrist and nerve pain?

Start with PCP: I would start with your primary care physician. He might want to order further testing or refer you to a neurologist or a neurosurgeon.

Neurologist : A neurologist should be able to diagnose and recommend you for treatment. Hand surgeons are usually either orthopedic or plastic surgeons.

Yes: One may see a physiatrist (also known as a physical medicine and rehabilitation physician). They are non-surgical bone, joint, nerve, muscle specialists. They may do a specialized test to evaluate the nerve/muscle (called and emg/ncs); it is like an ekg of the nerve/muscle.