I'm no expert at this sort of thing, but I recently found that I have a great grandfather in the 5th Michigan Calvary of Custer's Brigade. Various online sites had an Index for Union veterans which gave his his name, unit, and pension number, but no actual records. Finally, the service records and pension were ordered from the National Archives & Records Administration in Washington. At first the cost seemed a little high, but after receiving the records, it was well worth it for me. The NARA has a package deal if you order the pension and service records at the same time.

Others may have better methods, but this was my experience. Good luck.