Greetings from the UK ! There is a very nice illustration of officer's insignia on pages 144 - 145 in the Rebels & Yankees series entitled 'Commanders of the Civil War' publishesd by Salamander by William C Davis.
Rank structure in the Union Navy is as follows : Ensign; Master; Lieutenant; Lieutenant Commander; Commander; Captain; Commodore; Rear Admiral. There was no rank of Flag Officer, this was a generic term for a senior officer commanding a flotilla or squadron such as Silas Stringham at Hatteras Inlet or Francis Du Pont at Port Royal.
As to the Confederate Navy the rank structure was similar - Passed midshipman; Master; Lieutenant; Commander; Captain; however they did have the senior rank of Flag Officer.

Hope this helps

Regards

Peter Barratt
(author, Circle of Fire - The Story of the USS Susquehanna in the War of the Rebellion)