Is fibromyalgia something that a regular doctor can treat, or will I need to go to a pain clinic?

Anyone can treat. Anyone can treat it. Proper diagnosis may be best done by a rheumatologist or rehab medicine specialist or a neurologist.
Doctor interested? Any primary care doctor can successfully treat #fibromyalgia if he/she is interested in this fascinating disorder. Complementary and alternative therapies are often useful.
Depends on severity. Milder cases of fibromyalgia can be treated by family doctors. But, if patient has severe pain or complex pain symptoms then a consultation with a pain physician who manages fibromyalgia or a rheumatologist is more appropriate.