My GP diagnosed my FM and related problems. Rheumatologists (not sure of spelling) are the specialists who are usually the ones to diagnose FM. Not all doctors & specialists agree that FM/CFS are real conditions, so do your homework before seeing anyone. Check them out and see if they believe that FM/CFS are real. You can interview them before going any further with them. It's your money and your right to know if a Doctor is going to help you and work with you or just go through the motions to get your money!
As far as tests go, there is NO test to prove you have FM/CFS. It is a process of elimination, because the symtoms are very similar to so many other Auto immune conditions. It will depend on you and your Doctor and your Insurance Co. as to what tests will be need and when.
I am very sorry I could not give you a pat answer on this, but there is none. Good Luck on your search for a GOOD Doctor.